Simplify (3²)⁴×(5³)⁵: Complex Exponent Reduction Problem

Q: When do I multiply exponents versus add them?

Multiply exponents when you have a power raised to another power: $ (a^m)^n = a^{m \times n} $. Add exponents when multiplying same bases: $ a^m \times a^n = a^{m+n} $.

Q: Why can't I combine 3⁸ and 5¹⁵ further?

You can only combine terms with the same base. Since 3 and 5 are different bases, $ 3^8 \times 5^{15} $ is already in its simplest form.

Q: How do I remember the power of a power rule?

Think of it as repeated multiplication! $ (3^2)^4 $ means you're multiplying $ 3^2 $ four times: $ 3^2 \times 3^2 \times 3^2 \times 3^2 = 3^{2+2+2+2} = 3^8 $.

Q: What if I calculated (3²)⁴ as 9⁴ first?

That works too! $ 3^2 = 9 $, so $ (3^2)^4 = 9^4 $. But leaving it as $ 3^8 $ is usually the preferred simplified form since it shows the original base.

Q: Can I use a calculator for these large exponents?

For the final numerical answer, yes! But for algebra problems like this, teachers usually want the answer in exponential form like $ 3^8 \times 5^{15} $ rather than the huge number it equals.

Power of a Power with Multiple Bases

Reduce the following equation:

$\left(3^2\right)^4\times\left(5^3\right)^5=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 Let's start!

00:10 We'll use the formula for multiplying powers.

00:13 Any number, A, to the power of M, times the same number, A, to the power of N.

00:19 This gives us A to the power of M plus N.

00:23 We'll apply this formula in our exercise now.

00:27 First, match the numbers with the variables in the formula.

00:31 Keep the base the same and add the exponents together.

01:01 Use the same method for the second base.

01:25 And that's how we solve the question!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Reduce the following equation:

$\left(3^2\right)^4\times\left(5^3\right)^5=$

Step-by-step solution

To solve this problem, we'll employ the power of a power rule in exponents, which states that $(a^m)^n = a^{m \times n}$ .

Let's apply this rule to each part of the expression:

Step 1: Simplify $(3^2)^4$
According to the power of a power rule, this becomes $3^{2 \times 4} = 3^8$ .
Step 2: Simplify $(5^3)^5$
Similarly, apply the rule here to get $5^{3 \times 5} = 5^{15}$ .

After simplifying both parts, we multiply the results:

$3^8 \times 5^{15}$

Thus, the reduced expression is $\boxed{3^8 \times 5^{15}}$ .

Final Answer

$3^8\times5^{15}$

Key Points to Remember

Essential concepts to master this topic

Rule: When raising a power to a power, multiply the exponents
Technique: For $(3^2)^4$ , calculate 2×4 = 8, giving $3^8$
Check: Count operations: two power rules applied, bases stay separate ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of multiplying when using power of a power rule
Don't add the exponents like (3²)⁴ = 3²⁺⁴ = 3⁶! This treats it like multiplication rule instead of power rule and gives completely wrong results. Always multiply the exponents: (3²)⁴ = 3²ˣ⁴ = 3⁸.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

When do I multiply exponents versus add them?

Multiply exponents when you have a power raised to another power: $(a^m)^n = a^{m \times n}$ . Add exponents when multiplying same bases: $a^m \times a^n = a^{m+n}$ .

Why can't I combine 3⁸ and 5¹⁵ further?

You can only combine terms with the same base. Since 3 and 5 are different bases, $3^8 \times 5^{15}$ is already in its simplest form.

How do I remember the power of a power rule?

Think of it as repeated multiplication! $(3^2)^4$ means you're multiplying $3^2$ four times: $3^2 \times 3^2 \times 3^2 \times 3^2 = 3^{2+2+2+2} = 3^8$ .

What if I calculated (3²)⁴ as 9⁴ first?

That works too! $3^2 = 9$ , so $(3^2)^4 = 9^4$ . But leaving it as $3^8$ is usually the preferred simplified form since it shows the original base.

Can I use a calculator for these large exponents?

For the final numerical answer, yes! But for algebra problems like this, teachers usually want the answer in exponential form like $3^8 \times 5^{15}$ rather than the huge number it equals.

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