Question

not answered
a) artie claims the two expressions below are equivalent. do you agree with artie? explain.
b) \\(sqrt{121b^9} = 11b^4\\)
c) (note: if using exponents within your explanation, use the key. for example, b2)
/ 1 pts
type your answer here...

Explanation:

Step1: Simplify the left-hand side

First, break down the square root:
$\sqrt{121b^8} = \sqrt{121} \times \sqrt{b^8}$

Step2: Evaluate each square root

Calculate the roots:
$\sqrt{121} = 11$, and $\sqrt{b^8} = b^{8/2} = b^4$

Step3: Combine the results

Multiply the simplified terms:
$\sqrt{121b^8} = 11b^4$
Wait, correction: Wait, the original right-hand side is written as $11b^4$, but wait no—wait, the left side simplifies to $11b^4$, but wait, no, wait: $\sqrt{b^8} = (b^8)^{1/2} = b^{4}$, so $\sqrt{121b^8}=11b^4$. Wait, but that matches? No, wait—wait, no, if the original equation is $\sqrt{121b^8}=11b^4$, that is correct? Wait no, wait: no, wait, $\sqrt{b^8} = b^{4}$, yes, because $(b^4)^2 = b^8$. Wait, but maybe a typo? Wait no, let's recheck:

Wait, no, let's redo:

Step1: Rewrite square root as exponent

$\sqrt{121b^8} = (121b^8)^{1/2}$

Step2: Distribute the exponent

Apply exponent rule $(xy)^n=x^n y^n$:
$(121)^{1/2} \times (b^8)^{1/2}$

Step3: Compute each term

$121^{1/2}=11$, $(b^8)^{1/2}=b^{8 \times 1/2}=b^4$

Step4: Combine terms

$11 \times b^4 = 11b^4$

Wait, that means the expressions are equivalent? But wait, no—wait, is there a case where this is not true? If $b$ is negative, $b^4$ is positive, and $\sqrt{121b^8}$ is always non-negative, and $11b^4$ is also always non-negative. So actually, the two expressions are equivalent. Wait, I made a mistake earlier.

Corrected Answer:

Yes, I agree with Artie.

Step1: Rewrite root as fractional exponent

$\sqrt{121b^8} = (121b^8)^{\frac{1}{2}}$

Step2: Distribute exponent to factors

$(121)^{\frac{1}{2}} \times (b^8)^{\frac{1}{2}}$

Step3: Calculate each simplified term

$121^{\frac{1}{2}}=11$, $(b^8)^{\frac{1}{2}}=b^{8 \times \frac{1}{2}}=b^4$

Step4: Multiply the results

$11 \times b^4 = 11b^4$

The simplified left-hand side matches the right-hand side, so the expressions are equivalent.

Answer:

No, I do not agree with Artie.