7.3 涉及激进分子的溶解等同

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• 选择节涉及激进分子的溶解等同

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  涉及激进分子的溶解等同

  

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  Last week Sherri bought 324 square yards of sod to grass a square play area for the children of her day care. Now she has to fence the area to keep the children safe but does not know how many yards of fencing she needs to buy. All Sherri can figure out is that all the sides are the same length because the grassy area is a square.
  ::上周,雪莉购买了324平方码的草坪,为日托子女种植一个广场游戏区。 现在,她不得不围着这个区来保护孩子们的安全,但不知道她需要买多少码的围栏。 雪莉所能知道的只是,由于草地是一个广场,所有两边的长度都是一样的。

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  How can she determine how many yards of fencing to buy?
  ::她怎么能确定要买多少码的栅栏呢?

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  In this concept, you will learn to solve equations involving radicals.
  ::在这个概念中,你会学会解答 涉及激进分子的方程式。

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  Radicals
  ::激进

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  When you solve an equation you are trying to find the value for the variable that will make the equality statement true. The steps applied to solving an equation are inverse operations . To solve an equation involving radicals, inverse operations are used to solve for the variable.
  ::当您在解析一个方程式时, 您正试图找到使平等语句真实化的变量值。 用于解析等式的步骤是反向操作。 要解析一个涉及矩形的方程式, 则使用反向操作来解析变量 。

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  A radical involving the square root of a number can be evaluated by determining the square root of the number under the radical sign. If the radicand is a perfect square then its square will be the number which multiplied by itself twice will give the value of the radicand. For example the square root of 81 can be denoted by $\begin{align*}\sqrt{81}\end{align*}$ . What number times itself twice gives 81?
  ::涉及一个数字的平方根的激进分子可以通过在激进符号下确定数字的平方根来评价。如果弧线是一个完美的正方根,那么它的正方数将是它本身乘以两次的数值,它就会给弧线带来价值。例如,81的平方根可以用 81 来表示。如果弧线是完美的正方根,那么其正方数就会是它本身乘以两次的数值。比如,81 的平方根可以用 81 来表示。何方根乘以它本身乘以 81 ?

  $$\begin{align*}\begin{array}{rcl} 81 = 9 \times 9 &=& 9^2\\ \therefore \sqrt{81} &=& 9 \end{array}\end{align*}$$

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  Taking the square root of a number is the inverse operation of squaring and vice versa.
  ::取一个数字的平方根是对角的反作用,反之亦然。

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  Let’s look at an equation involving radicals.
  ::让我们来看看一个涉及激进分子的方程式。

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  $$\begin{align*}x^2 = 121\end{align*}$$
  ::x2=121x2=121

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  The variable ‘ $\begin{align*}x\end{align*}$ ’ is squared and its value is 121. To solve this equation the value of needs to be determined. The inverse operation of squaring is taking the square root. Remember, whatever operation is applied to one side of the equation must also be applied to the other side.
  ::变量 `x' 是平方的, 其值是 121 。 要解决这个方程, 需要确定数值。 方程的反向操作取平方根。 记住, 无论对方程的一边应用何种操作, 也必须对另一方应用 。

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  First, take the square root of both sides of the equation.
  ::首先,选择方程式两侧的平方根。

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  $$\begin{align*}\begin{array}{rcl} x^2 &=& 121\\ \sqrt{x^2} &=& \sqrt{121}\\ x &=& 11 \end{array}\end{align*}$$
  ::x2 = 121x2 121x= 11

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  Next, verify the answer by substituting the value for ‘ $\begin{align*}x\end{align*}$ ’ into the original equation.
  ::下一步,验证答案, 将 `x ' 的值替换为原始方程式 。

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  $$\begin{align*}\begin{array}{rcl} x^2 &=& 121\\ (11)^2 &=& 121 \end{array}\end{align*}$$
  ::x2=121( 11)2=121

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  Then, perform any indicated operations.
  ::然后,执行任何指定的行动。

  $$\begin{align*}\begin{array}{rcl} 11 \times 11 &=& 121\\ 121 &=& 121 \end{array}\end{align*}$$

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  The answer is 11.
  ::答案是11个

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  The value of 11 made the equality statement true – both sides of the equation are the same.
  ::11的数值让平等声明成为事实, 等式的两面都是一样的。

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  Let’s look at one more.
  ::让我们再看一眼。

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  Solve the following equation involving radicals:
  ::解决以下涉及激进分子的方程式:

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  $$\begin{align*}\sqrt{x+2}=6\end{align*}$$
  ::x+2=6

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  Notice the left side of the equation is a radical. The inverse operation of taking the square root is squaring.
  ::注意方程的左侧是激进的。 取取平方根的反作用是激烈的。

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  Remember, the square of the square root of anything is the anything. In other words
  ::记住,平方根的正方根是任何东西。换句话说,

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  $$\begin{align*}\left(\sqrt{\text{anything}} \right)^2 = \text{anything}.\end{align*}$$
  :任何东西)2 任何东西。

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  First, square both sides of equation.
  ::首先,两个方程式的两面都对齐。

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  $$\begin{align*}\begin{array}{rcl} \sqrt{x+2} &=& 6\\ \left(\sqrt{x+2} \right)^2 &=& (6)^2 \end{array}\end{align*}$$
  ::x+2=6(xx+2)2=(6)2

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  Next, perform any indicated operations and simplify the equation.
  ::下一步,执行任何指明的行动并简化方程。

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  $$\begin{align*}\begin{array}{rcl} \left(\sqrt{x+2} \right)^2 &=& x+2\\ (6)^2 = 6 \times 6 &=& 36\\ x+2 &=& 36 \end{array}\end{align*}$$
  :xx+2)2=x+2(6)2=6x6=36x+2=36

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  Then, subtract 2 from both sides of the equation to solve the equation for ‘ $\begin{align*}x\end{align*}$ ’.
  ::然后从方程的两侧减去2, 以解答“ x” 的方程 。

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  $$\begin{align*}\begin{array}{rcl} x+2 &=& 36\\ x+2-2 &=& 36-2\\ x &=& 34 \end{array}\end{align*}$$
  ::x+2=36x+2-2=36-2x=34

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  Next, verify the answer by substituting the value for ‘ $\begin{align*}x\end{align*}$ ’ into the original equation.
  ::下一步,验证答案, 将 `x ' 的值替换为原始方程式 。

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  $$\begin{align*}\begin{array}{rcl} \sqrt{x+2} &=& 6\\ \sqrt{{\color{blue}34}+2} &=& 6 \end{array}\end{align*}$$
  ::x+2=634+2=6

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  Then, perform any indicated operations and simplify the equation.
  ::然后,执行任何指示的行动并简化方程。

  $$\begin{align*}\begin{array}{rcl} \sqrt{{\color{blue}34}+2} &=& 6\\ \sqrt{36} &=& 6\\ 36 = 6 \times 6 &=& 6^2\\ 6 &=& 6 \end{array}\end{align*}$$

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  The answer is 34.
  ::答案是34岁

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  The value of 34 made the equation true.
  ::34的值使方程成为事实。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were given a problem about Sherri and the fence for the grassy square. She needs to figure out the perimeter of the square.
  
  First, write an equation for the area of the grassy square.
  ::早些时候,有人给了你一个关于Sherri和草原广场围栏的问题。 她需要弄清楚广场的周界。 首先,写一个草原广场面积的方程。

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  $$\begin{align*}A = s^2\end{align*}$$
  ::A=s2 阿=2

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  Such that $\begin{align*}A\end{align*}$  is the area and $\begin{align*}s\end{align*}$  is the length of any side of the square.
  ::A是面积,s是方形任何一侧的长度。

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  Next, fill in the value for the area.
  ::下一步,填充此区域的值 。

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  $$\begin{align*}\begin{array}{rcl} A &=& s^2\\ 324 &=& s^2 \end{array}\end{align*}$$
  ::A=s2324=s2

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  Next, solve the equation for ‘ $\begin{align*}s\end{align*}$ ’ by taking the square root of both sides of the radical equation.
  ::以激进方程式两边的平方根来解决`s ' 的方程式。

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  $$\begin{align*}\begin{array}{rcl} 324 &=& s^2\\ \sqrt{324} &=& \sqrt{s^2}\\ 324 &=& 18 \times 18\\ s^2 &=& s \times s\\ 18 &=& s \end{array}\end{align*}$$
  ::324=s2324s2324=18×18s2=sxs18=s

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  The answer is 18.
  ::答案是18岁

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  The length of each side of the square is 18 yds.
  ::方形每一侧的长度为18 yds。

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  Next, write an equation for finding the perimeter of the square. The perimeter is the distance all around the grassy area.
  ::接下来,写一个方程式以寻找广场的周界。 周界是草地周围的距离。

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  $$\begin{align*}P = 4s\end{align*}$$
  ::P=4s

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  Such that ‘ $\begin{align*}P\end{align*}$ ’ is the perimeter and ‘ $\begin{align*}s\end{align*}$ ’ is the length of one side of the square.
  ::因此`P ' 是周边, 's ' 是广场一面的长度。

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  Next, fill in the value for ‘ $\begin{align*}s\end{align*}$ ’ and perform the indicated operation.
  ::接下来填上`s ' 的值,并进行指定的操作。

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  $$\begin{align*}\begin{array}{rcl} P &=& 4s\\ P &=& 4(18)\\ P &=& 72 \end{array}\end{align*}$$
  ::P=4sP=4(18)P=72

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  The answer is 72.
  ::答案是72。

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  Sherri needs to buy 72 yards of fencing.
  ::雪莉需要买72码的栅栏

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  Example 2
  ::例2

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  Solve the following equation involving radicals to the nearest tenth.
  ::解决下个方程, 包括最接近的十分之一的立方体。

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  $$\begin{align*}x^2 = 26\end{align*}$$
  ::x2=26

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  First, the opposite operation of squaring is taking the square root. Take the square root of both sides of the equation.
  ::首先,相反的对角操作取平方根。 取平方法两侧的平方根 。

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  $$\begin{align*}\begin{array}{rcl} x^2 &=& 26\\ \sqrt{x^2} &=& \sqrt{26} \end{array}\end{align*}$$
  ::x2=26x226

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  Notice that the number 26 is not a perfect square. Use the TI calculator to find the square root of 26.
  ::注意数字 26 不是一个完美的正方形。 使用 TI 计算器找到 26 的平方根 。

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  Next, on the calculator press 2 ^nd $\begin{align*}x^2 \quad 2 \quad 6\end{align*}$    enter.
  ::下一步,在计算器按 2ndx226 输入 。

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  Then, round the value shown on the screen to one place after the decimal.
  ::然后,将屏幕上显示的值绕到小数点后的一个位置。

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  $$\begin{align*}\begin{array}{rcl} \sqrt{x^2} &=& \sqrt{26}\\ x &=& 5.099019514\\ x &=& 5.1 \end{array}\end{align*}$$
  ::x226x=5.99019514x=5.1

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  The answer is 5.1
  ::答案是5.1

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  Example 3
  ::例3

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  Solve the following radical equation for the variable ‘ $\begin{align*}m\end{align*}$ ’.
  ::解决变量`m'的以下激进方程。

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  $$\begin{align*}m^2 = 144\end{align*}$$
  ::平方米=144

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  First, take the square root of both sides of the equation.
  ::首先,选择方程式两侧的平方根。

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  $$\begin{align*}\begin{array}{rcl} m^2 &=& 144\\ \sqrt{m^2} &=& \sqrt{144} \end{array}\end{align*}$$
  ::平方公尺=144平方公尺

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  Next, simplify both sides of the equation.
  ::接下来,简化方程式的两边

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  $$\begin{align*}\begin{array}{rcl} \sqrt{m^2} &=& \sqrt{144}\\ m^2 &=& m \times m\\ 144 &=& 12 \times 12\\ m &=& 12 \end{array}\end{align*}$$
  ::m2144m2=m×m144=12×12m=12

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  The answer is 12.
  ::答案是12岁

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  Example 4
  ::例4

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  Solve the following radical equation for the variable ‘ $\begin{align*}a\end{align*}$ ’.
  ::解决变量“a”的以下激进方程。

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  $$\begin{align*}\sqrt{a-8} = 7\end{align*}$$
  ::a-8=7

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  First, apply the inverse operation of taking the square root to both sides of the equation.
  ::首先,将平方根对等方程的两侧应用反向操作。

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  $$\begin{align*}\begin{array}{rcl} \sqrt{a-8} &=& 7\\ \left(\sqrt{a-8} \right)^2 &=& (7)^2 \end{array}\end{align*}$$
  ::a-8=7(a-8)2=(7)2

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  Next, perform any indicated operations and simplify the equation.
  ::下一步,执行任何指明的行动并简化方程。

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  $$\begin{align*}\begin{array}{rcl} \left(\sqrt{a-8} \right)^2 &=& (7)^2\\ a-8 &=& 7 \times 7\\ a-8 &=& 49 \end{array}\end{align*}$$
  :a-8)2=(7)2a-8=7×7a-8=8=49

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  Then, add eight to both sides of the equation to solve for the variable ‘ $\begin{align*}a\end{align*}$ ’.
  ::然后,在等式的两边加八,以解决变量“a”。

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  $$\begin{align*}\begin{array}{rcl} a-8 &=& 49\\ a-8+8 &=& 49+8\\ a &=& 57 \end{array}\end{align*}$$
  ::a-8=49a-8+8=49+8a=57

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  Next, verify the answer by substituting the value for ‘ $\begin{align*}a\end{align*}$ ’ into the original equation.
  ::接下来,验证答案,将“a”值替换为原始方程式。

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  $$\begin{align*}\begin{array}{rcl} \sqrt{a-8} &=& 7\\ \sqrt{{\color{blue}57}-8} &=& 7 \end{array}\end{align*}$$
  ::a-8=7-57-8=7

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  Then, perform any indicated operations and simplify the equation.
  ::然后,执行任何指示的行动并简化方程。

  $$\begin{align*}\begin{array}{rcl} \sqrt{{\color{blue}57}-8} &=& 7\\ \sqrt{49} &=& 7\\ 49 &=& 7 \times 7\\ 7 &=& 7 \end{array}\end{align*}$$

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  The answer is 57.
  ::答案是57。

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  Example 5
  ::例5

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  Solve the following equation for the variable.
  ::解决变量的以下方程式 。

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  $$\begin{align*}x^3+4 = 31\end{align*}$$
  ::x3+4=31

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  First, isolate the variable by subtracting 4 from both sides of the equation.
  ::首先,从方程的两侧减去4,将变量分离出来。

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  $$\begin{align*}\begin{array}{rcl} x^3+4 &=& 31\\ x^3+4-4 &=& 31-4 \end{array}\end{align*}$$
  ::x3+4=31x3+4-4=31-4

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  Next, simplify both sides of the equation.
  ::接下来,简化方程式的两边

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  $$\begin{align*}\begin{array}{rcl} x^3+4-4 &=& 31-4\\ x^3 &=& 27 \end{array}\end{align*}$$
  ::x3+4-4=31-4x3=27

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  Next, perform the inverse operation of cubing – taking the cube root , on both sides of the equation.
  ::接下来,在方程的两侧 进行相反的抱抱操作, 取下立方根。

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  $$\begin{align*}\begin{array}{rcl} x^3 &=& 27\\ \sqrt[3]{x^3} &=& \sqrt[3]{27} \end{array}\end{align*}$$
  ::x3 = 273x3= 3________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  Then, perform the indicated operations.
  ::然后,执行指示的行动。

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  $$\begin{align*}\begin{array}{rcl} \sqrt[3]{x^3} &=& \sqrt[3]{27}\\ \sqrt[3]{x^3} &=& x \times x \times x\\ \sqrt[3]{27} &=& 3 \times 3 \times 3\\ x &=& 3 \end{array}\end{align*}$$  
  ::3x3=3273×3=xxxxx3x3}27=3×3x3x3x3=3

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  The answer is 3.
  ::答案是3

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  Review
  ::回顾

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  Solve each equation involving radical expressions.
  ::解决每个涉及激进表达式的方程式。

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  1. $\begin{align*}x^2 = 121\end{align*}$   
     ::x2=121x2=121
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  2. $\begin{align*}x^2 = 144\end{align*}$
     ::x2=144x2=144
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  3. $\begin{align*}x^2 = 64\end{align*}$
     ::x2=64
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  4. $\begin{align*}x^2 = 169\end{align*}$
     ::x2=169
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  5. $\begin{align*}x^2 = 16\end{align*}$
     ::x2=16
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  6. $\begin{align*}x^3 = 64\end{align*}$
     ::x3=64
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  7. $\begin{align*}x^3 = 27\end{align*}$  
     ::x3=27x3=27
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  8. $\begin{align*}x^2+3 = 147\end{align*}$
     ::x2+3=147
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  9. $\begin{align*}x^2-2 = 23\end{align*}$
     ::x2-2=23
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  10. $\begin{align*}x^2+5 = 30\end{align*}$
      ::x2+5=30
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  11. $\begin{align*}x^3+4 = 68\end{align*}$
      ::x3+4=68
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  12. $\begin{align*}x^3+10 = 135\end{align*}$
      ::x3+10=135 x3+10=135
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  13. $\begin{align*}x^2-4 = 21\end{align*}$
      ::x2 - 4=21
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  14. $\begin{align*}x^2-6 = 30\end{align*}$
      ::x2 - 6=30
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  15. $\begin{align*}x^2-12 = 37\end{align*}$
      ::x2 - 12=37

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  Review (Answers)
  ::回顾(答复)

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  Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
  ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。

   

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  Resources
  ::资源