Chapter 8: Q.515 (page 853)

In the following exercises, simplify
$(a) 6^{\frac{5}{2}} \cdot 6^{\frac{1}{2}} (b) {(b^{15})}^{\frac{3}{5}} (c) \frac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}}$

Short Answer

Expert verified

Simplifying,

$(a) 6^{\frac{5}{2}} \cdot 6^{\frac{1}{2}} = 216$

$(b) {(b^{15})}^{\frac{3}{5}} = b^{9}$

$(c) \frac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}} = \frac{1}{w}$

Step by step solution

Step 1. Given information

We have given,

$(a) 6^{\frac{5}{2}} \cdot 6^{\frac{1}{2}} (b) {(b^{15})}^{\frac{3}{5}} (c) \frac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}}$

Step 2. Concept

We will use the,

Product rule of exponent:

$a^{b} \cdot a^{c} = a^{b + c}$

Power rule of exponent:

${(a^{b})}^{c} = a^{b \cdot c}$

Quotient rule,

$\frac{x^{a}}{x^{b}} = x^{a - b}$

Negative exponent rule:

$a^{- n} = \frac{1}{a^{n}}$

Part (a) Step 1. Explanation

We have given,

$6^{\frac{5}{2}} \cdot 6^{\frac{1}{2}}$

Using product rule of the exponent,

$6^{\frac{5}{2}} \cdot 6^{\frac{1}{2}} = 6 ({^{\frac{5}{2} +}}^{\frac{1}{2}})$

Simplifying,

$6^{\frac{5}{2}} \cdot 6^{\frac{1}{2}} = 6^{3}$

$\Rightarrow 6^{\frac{5}{2}} \cdot 6^{\frac{1}{2}} = 216$

Part (a) Step 2. Conclusion

Hence, value of $6^{\frac{5}{2}} \cdot 6^{\frac{1}{2}}$ is 216.

Part (b) Step 1. Explanation

We have,

${(b^{15})}^{\frac{3}{5}}$

Using power rule of the exponent,

role="math" localid="1645078066918" ${(b^{15})}^{\frac{3}{5}} = b^{15 \cdot \frac{3}{5}}$

Simplifying,

${(b^{15})}^{\frac{3}{5}} = b^{9}$

Part (b) Step 2. Conclusion

Hence, value of ${(b^{15})}^{\frac{3}{5}}$ is $b^{9}$ .

Part (c) Step 1. Explanation

We have,

$\frac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}}$ ,

Using quotient rule,

$\frac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}} = w (^{\frac{2}{7} - \frac{9}{7}})$

Simplifying,

$\frac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}} = w (^{\frac{2 - 9}{7}})$

$\Rightarrow \frac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}} = w^{- 1}$

Using negative exponent rule,

$\frac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}} = \frac{1}{w}$

Part (c) Step 2. Conclusion

Hence, value of, $\frac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}}$ is $\frac{1}{w}$ .

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In the following exercises, simplify
$(a) 6^{\frac{5}{2}} \cdot 6^{\frac{1}{2}} (b) {(b^{15})}^{\frac{3}{5}} (c) \frac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Concept

Part (a) Step 1. Explanation

Part (a) Step 2. Conclusion

Part (b) Step 1. Explanation

Part (b) Step 2. Conclusion

Part (c) Step 1. Explanation

Part (c) Step 2. Conclusion

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Short Answer

Step by step solution

Step 1. Given information

Step 2. Concept

Part (a) Step 1. Explanation

Part (a) Step 2. Conclusion

Part (b) Step 1. Explanation

Part (b) Step 2. Conclusion

Part (c) Step 1. Explanation

Part (c) Step 2. Conclusion

One App. One Place for Learning.

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In the following exercises, simplify
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