Simplify ((by)^8)^9: Evaluating Compound Exponent Expressions

Q: Why do I multiply the exponents instead of adding them?

The power of a power rule says $ (x^m)^n = x^{m \times n} $. Think of it this way: $ ((by)^8)^9 $ means you're multiplying $ (by)^8 $ by itself 9 times, which creates 8 × 9 = 72 total factors of $ (by) $.

Q: What's the difference between this and multiplying terms with exponents?

Great question! When you have $ x^8 \times x^9 $, you add exponents to get $ x^{17} $. But when you have $ (x^8)^9 $, you multiply exponents to get $ x^{72} $. The parentheses make all the difference!

Q: How can I remember which rule to use?

Look for parentheses! If you see something like $ (expression)^{power} $, multiply the exponents inside and outside. If there are no parentheses around a power, like $ x^8 \times x^9 $, add the exponents.

Q: Does it matter that we have two variables (b and y)?

No! The rule works the same way. Treat $ (by) $ as a single unit or base. Whether it's $ ((by)^8)^9 $ or $ (x^8)^9 $, you still multiply: 8 × 9 = 72.

Q: What if I got a different answer like 9/8?

Getting $ \frac{9}{8} $ suggests you might have tried division instead of multiplication. Remember: power of a power always means multiply. There's no division involved in $ ((by)^8)^9 $.

Step-by-step video solution

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Step-by-step written solution

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1

Understand the problem

Insert the corresponding expression:

$\left(\left(by\right)^8\right)^9=$

2

Step-by-step solution

To solve this problem, we'll apply the power of a power rule for exponents. The rule states that if you have an expression of the form $(x^m)^n$ , it simplifies to $x^{m \cdot n}$ .

Step 1: Identify the given expression: $\left(\left(by\right)^8\right)^9$ .
Step 2: Apply the power of a power rule by multiplying the exponents.

Let's work through the solution:
Step 1: We start with $\left(\left(by\right)^8\right)^9$ . Here, $\left(by\right)$ is considered as a single base.
Step 2: Apply the power of a power rule: $(by)^{8 \cdot 9}$ .
Step 3: Calculate the exponent multiplication: $8 \times 9 = 72$ .

Therefore, the simplified expression is $(b \times y)^{72}$ .

Analyzing the choices provided:

Choice 1: $(b \times y)^{17}$ - Incorrect because the exponents should multiply to 72.
Choice 2: $(b \times y)^1$ - Incorrect because it does not reflect the multiplication of exponents.
Choice 3: $(b \times y)^{\frac{9}{8}}$ - Incorrect, involves incorrect operations on the exponents.
Choice 4: $(b \times y)^{72}$ - Correct as it correctly applies the power of a power rule.

Thus, the correct answer is Choice 4: $\left(b \times y\right)^{72}$ .

3

Final Answer

$\left(b\times y\right)^{72}$

Key Points to Remember

Essential concepts to master this topic

Power of a Power Rule: When raising a power to another power, multiply the exponents
Technique: $((by)^8)^9 = (by)^{8 \times 9} = (by)^{72}$
Check: Verify exponent multiplication: 8 × 9 = 72, not addition (8 + 9 = 17) ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of multiplying them
Don't add 8 + 9 = 17 to get $(by)^{17}$ ! This gives a completely wrong result because you're applying the wrong rule. Always multiply exponents when raising a power to another power: $((by)^8)^9 = (by)^{8 \times 9} = (by)^{72}$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I multiply the exponents instead of adding them?

+

The power of a power rule says $(x^m)^n = x^{m \times n}$ . Think of it this way: $((by)^8)^9$ means you're multiplying $(by)^8$ by itself 9 times, which creates 8 × 9 = 72 total factors of $(by)$ .

What's the difference between this and multiplying terms with exponents?

+

Great question! When you have $x^8 \times x^9$ , you add exponents to get $x^{17}$ . But when you have $(x^8)^9$ , you multiply exponents to get $x^{72}$ . The parentheses make all the difference!

How can I remember which rule to use?

+

Look for parentheses! If you see something like $(expression)^{power}$ , multiply the exponents inside and outside. If there are no parentheses around a power, like $x^8 \times x^9$ , add the exponents.

Does it matter that we have two variables (b and y)?

+

No! The rule works the same way. Treat $(by)$ as a single unit or base. Whether it's $((by)^8)^9$ or $(x^8)^9$ , you still multiply: 8 × 9 = 72.

What if I got a different answer like 9/8?

+

Getting $\frac{9}{8}$ suggests you might have tried division instead of multiplication. Remember: power of a power always means multiply. There's no division involved in $((by)^8)^9$ .

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Simplify ((by)^8)^9: Evaluating Compound Exponent Expressions

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