Question

please, show your work!!!
i can use both radicals and exponents to represent numbers.

1. $6^2 = 36$ and $6^3 = 216$

a. find the value of each variable that makes the equation true.
i. $36^a = 6$ 1 point ____
ii. $36^b = 216$ 1 point ____
b. use the value you found for $b$ to write the equation $36^b = 216$ using a radical and whole number power: 1 point ____
c. write the value of $9^{-\frac{1}{2}}$ using no exponents or radicals. 1 point ____
1
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Explanation:

Step1: Rewrite 36 as $6^2$

$36^a = (6^2)^a = 6^{2a}$

Step2: Set exponent equal to 1

$2a = 1 \implies a = \frac{1}{2}$

Step3: Rewrite 36 and 216 as powers of 6

$36^b = (6^2)^b = 6^{2b}$, $216 = 6^3$

Step4: Set exponents equal

$2b = 3 \implies b = \frac{3}{2}$

Step5: Rewrite $36^b$ with radical

$36^{\frac{3}{2}} = (36^{\frac{1}{2}})^3 = (\sqrt{36})^3$

Step6: Rewrite $9^{-\frac{1}{2}}$ with positive exponent

$9^{-\frac{1}{2}} = \frac{1}{9^{\frac{1}{2}}}$

Step7: Simplify the denominator

$9^{\frac{1}{2}} = \sqrt{9} = 3$, so $\frac{1}{3}$

Answer:

a. i. $\boldsymbol{\frac{1}{2}}$
a. ii. $\boldsymbol{\frac{3}{2}}$
b. $\boldsymbol{(\sqrt{36})^3 = 216}$
c. $\boldsymbol{\frac{1}{3}}$