Simplify the Expression: 5^(y+1) × 3^(y+1) Product Rule Challenge

Q: How do I know when I can combine bases?

You can only combine bases when the exponents are exactly the same. In this problem, both terms have the exponent $ y+1 $, so we can use the rule $ a^m \times b^m = (a \times b)^m $.

Q: What if the exponents were different?

If the exponents were different, you cannot combine the bases! For example, $ 5^{y+1} \times 3^{y+2} $ cannot be simplified using this rule because the exponents don't match.

Q: Do I multiply the exponents together?

No! When combining bases with the same exponent, you keep the same exponent. Only multiply exponents when raising a power to a power, like $ (5^2)^3 = 5^6 $.

Q: Can I simplify 5 × 3 in the final answer?

The question asks for the corresponding expression, so $ (5 \times 3)^{y+1} $ is the correct form. You could write it as $ 15^{y+1} $, but the factored form shows the original bases clearly.

Q: What's the difference between this and the product rule for same bases?

Great question! For same bases: $ 5^a \times 5^b = 5^{a+b} $ (add exponents). For same exponents: $ 5^a \times 3^a = (5 \times 3)^a $ (combine bases). Always check what's the same first!

Power Rules with Same Exponents

Insert the corresponding expression:

$5^{y+1}\times3^{y+1}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:09 Let's simplify the problem together.

00:13 According to the rules of exponents, when a product is raised to a power, like N,

00:20 it means each factor is also raised to that power N.

00:24 We'll use this rule in our example now.

00:28 Notice how the exponent N has an addition sign.

00:32 Remember, each factor is raised to the same power, N.

00:37 And there you have it! That's the solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$5^{y+1}\times3^{y+1}=$

Step-by-step solution

To solve this problem, we'll use the "Power of a Product" rule.

We begin with the expression $5^{y+1} \times 3^{y+1}$ .

Notice that both terms share the same exponent, $y+1$ .

According to the Power of a Product rule, $a^m \times b^m = (a \times b)^m$ . This means we can combine $5^{y+1} \times 3^{y+1}$ into a single expression.

Let's apply the formula:

Identify each base: $a = 5$ and $b = 3$ .
The shared exponent is $m = y+1$ .
Substitute into the formula: $(5 \times 3)^{y+1}$ .

Therefore, the corresponding expression is $\left(5 \times 3\right)^{y+1}$ .

Final Answer

$\left(5\times3\right)^{y+1}$

Key Points to Remember

Essential concepts to master this topic

Rule: When bases differ but exponents match, combine bases: $a^m \times b^m = (a \times b)^m$
Technique: Check exponents first - both have $y+1$ , so combine: $5^{y+1} \times 3^{y+1} = (5 \times 3)^{y+1}$
Check: Verify by expanding back: $(5 \times 3)^{y+1} = 15^{y+1} = 5^{y+1} \times 3^{y+1}$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of combining bases
Don't add the exponents to get $(5 \times 3)^{2y+2}$ ! This rule only applies when multiplying powers with the same base. Always check if exponents are identical first, then combine bases with that same exponent.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

How do I know when I can combine bases?

You can only combine bases when the exponents are exactly the same. In this problem, both terms have the exponent $y+1$ , so we can use the rule $a^m \times b^m = (a \times b)^m$ .

What if the exponents were different?

If the exponents were different, you cannot combine the bases! For example, $5^{y+1} \times 3^{y+2}$ cannot be simplified using this rule because the exponents don't match.

Do I multiply the exponents together?

No! When combining bases with the same exponent, you keep the same exponent. Only multiply exponents when raising a power to a power, like $(5^2)^3 = 5^6$ .

Can I simplify 5 × 3 in the final answer?

The question asks for the corresponding expression, so $(5 \times 3)^{y+1}$ is the correct form. You could write it as $15^{y+1}$ , but the factored form shows the original bases clearly.

What's the difference between this and the product rule for same bases?

Great question! For same bases: $5^a \times 5^b = 5^{a+b}$ (add exponents). For same exponents: $5^a \times 3^a = (5 \times 3)^a$ (combine bases). Always check what's the same first!

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