Simplify (15×2)¹⁷ ÷ (2×15)¹³: Power Rules in Action

Q: Why are (15×2) and (2×15) considered the same base?

Because of the commutative property of multiplication! The order of factors doesn't change the product: $ 15 \times 2 = 2 \times 15 = 30 $. They represent the same base value.

Q: What if the bases were actually different?

If the bases were truly different, you couldn't use the quotient rule to simplify. You'd need to calculate each power separately first, then divide the results.

Q: Should I calculate the final answer as 30⁴?

The question asks for the expression form, not the numerical value. Keep it as $ (15 \times 2)^{17-13} $ to show your understanding of the quotient rule.

Q: How do I remember the quotient rule?

Think: "Same base, subtract the powers" - $ \frac{a^m}{a^n} = a^{m-n} $. When dividing, you're removing the bottom power from the top power.

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

The product rule adds exponents: $ a^m \times a^n = a^{m+n} $. The quotient rule subtracts exponents: $ \frac{a^m}{a^n} = a^{m-n} $. Remember: multiply = add, divide = subtract!

Quotient Rule with Same Base Expressions

Insert the corresponding expression:

$\frac{\left(15\times2\right)^{17}}{\left(2\times15\right)^{13}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:03 In multiplication, the order of factors doesn't matter

00:09 We'll use this formula in our exercise and reverse the order of factors

00:16 We'll use the formula for dividing powers

00:19 Any number (A) to the power of (N) divided by the same base (A) to the power of (M)

00:22 equals the number (A) to the power of the difference of exponents (M-N)

00:25 We'll use this formula in our exercise

00:29 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{\left(15\times2\right)^{17}}{\left(2\times15\right)^{13}}=$

Step-by-step solution

The given expression is $\frac{\left(15\times2\right)^{17}}{\left(2\times15\right)^{13}}$ . To simplify
using the rule of exponents known as the Power of a Quotient Rule, which states

When you divide like bases you subtract the exponents:

$\frac{a^m}{a^n} = a^{m-n}$ .

First, notice that both the numerator and denominator have the base $15 \times 2$ . Therefore, we can simplify by subtracting the exponents in the numerator and the denominator:

Numerator's exponent: $17$
Denominator's exponent: $13$

We apply the quotient rule:

(15 \times 2)^{17-13}

.

As a result, the simplified expression is

(15 \times 2)^4

.
Therefore, the correct answer which represents the expression using the Power of a Quotient Rule is

\left(15\times2\right)^{17-13}

because it encapsulates the subtraction of exponents without computing the final exponent

4

.

This allows you to keep the expression in its most simplistic exponential form.

Final Answer

$\left(15\times2\right)^{17-13}$

Key Points to Remember

Essential concepts to master this topic

Quotient Rule: When dividing same bases, subtract the exponents
Technique: Recognize that (15×2) and (2×15) are identical bases
Check: Verify base equality: 15×2 = 2×15 = 30 ✓

Common Mistakes

Avoid these frequent errors

Not recognizing identical bases due to different order
Don't think (15×2) and (2×15) are different bases = wrong application of rules! Order doesn't matter in multiplication, so these are identical. Always identify when bases are the same regardless of factor arrangement.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why are (15×2) and (2×15) considered the same base?

Because of the commutative property of multiplication! The order of factors doesn't change the product: $15 \times 2 = 2 \times 15 = 30$ . They represent the same base value.

What if the bases were actually different?

If the bases were truly different, you couldn't use the quotient rule to simplify. You'd need to calculate each power separately first, then divide the results.

Should I calculate the final answer as 30⁴?

The question asks for the expression form, not the numerical value. Keep it as $(15 \times 2)^{17-13}$ to show your understanding of the quotient rule.

How do I remember the quotient rule?

Think: "Same base, subtract the powers" - $\frac{a^m}{a^n} = a^{m-n}$ . When dividing, you're removing the bottom power from the top power.

What's the difference between this and the product rule?

The product rule adds exponents: $a^m \times a^n = a^{m+n}$ . The quotient rule subtracts exponents: $\frac{a^m}{a^n} = a^{m-n}$ . Remember: multiply = add, divide = subtract!

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