Simplify the Expression: (2a)⁵ ÷ (2a)³ Using Laws of Exponents

Q: Why do we subtract exponents when dividing?

Think of it this way: $ \frac{x^5}{x^3} = \frac{x \cdot x \cdot x \cdot x \cdot x}{x \cdot x \cdot x} $. When you cancel out the common factors, you're left with x², which is x^(5-3)!

Q: What if the bases look different but are actually the same?

Great observation! In this problem, $ (2 \times a) $ and $ (2a) $ are the same base - just written differently. Always identify the base carefully before applying the quotient rule.

Q: Can I simplify (2a)^(5-3) further?

Yes! $ (2a)^{5-3} = (2a)^2 = 4a^2 $. But in this question, they want the intermediate step showing the subtraction of exponents first.

Q: What happens if the bottom exponent is bigger than the top?

You still subtract! For example, $ \frac{x^2}{x^5} = x^{2-5} = x^{-3} = \frac{1}{x^3} $. The negative exponent tells you the result goes in the denominator.

Q: Does this rule work with any base?

Absolutely! The quotient rule $ \frac{a^m}{a^n} = a^{m-n} $ works for any base (numbers, variables, or expressions) as long as the base is exactly the same in both numerator and denominator.

Quotient Rule with Same Base Expressions

Insert the corresponding expression:

$\frac{\left(2\times a\right)^5}{\left(2\times a\right)^3}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:02 We'll use the formula for dividing powers

00:04 Any number (A) to the power of (N) divided by the same base (A) to the power of (M)

00:07 equals number (A) to the power of the difference of exponents (M-N)

00:09 We'll use this formula in our exercise

00:11 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:

$\frac{\left(2\times a\right)^5}{\left(2\times a\right)^3}=$

Step-by-step solution

To solve the given expression, we apply the Power of a Quotient Rule for Exponents. This rule tells us that if we have an expression of the form $\frac{b^m}{b^n}$ , it simplifies to $b^{m-n}$ .

Given the expression $\frac{(2\times a)^5}{(2\times a)^3}$ , we can identify it with the rule as follows. Here, the base $(2\times a)$ is the same in both the numerator and the denominator, with exponents 5 and 3 respectively.

According to the rule, we subtract the exponent in the denominator from the exponent in the numerator, which results in $(2\times a)^{5-3}$ .

This simplifies to $(2\times a)^2$ , but based on the way the answer is expected to be expressed, we stick with $(2\times a)^{5-3}$ .

Thus, the solution to the question is: $(2\times a)^{5-3}$

Final Answer

$\left(2\times a\right)^{5-3}$

Key Points to Remember

Essential concepts to master this topic

Quotient Rule: When dividing same bases, subtract the exponents
Technique: $\frac{(2a)^5}{(2a)^3} = (2a)^{5-3}$ by keeping base unchanged
Check: Verify (2a)² = 4a² when expanded correctly ✓

Common Mistakes

Avoid these frequent errors

Adding or multiplying exponents instead of subtracting
Don't add exponents (5+3=8) or multiply them (5×3=15) when dividing = completely wrong operations! Division of same bases requires subtraction, not addition or multiplication. Always subtract the bottom exponent from the top exponent.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do we subtract exponents when dividing?

Think of it this way: $\frac{x^5}{x^3} = \frac{x \cdot x \cdot x \cdot x \cdot x}{x \cdot x \cdot x}$ . When you cancel out the common factors, you're left with x², which is x^(5-3)!

What if the bases look different but are actually the same?

Great observation! In this problem, $(2 \times a)$ and $(2a)$ are the same base - just written differently. Always identify the base carefully before applying the quotient rule.

Can I simplify (2a)^(5-3) further?

Yes! $(2a)^{5-3} = (2a)^2 = 4a^2$ . But in this question, they want the intermediate step showing the subtraction of exponents first.

What happens if the bottom exponent is bigger than the top?

You still subtract! For example, $\frac{x^2}{x^5} = x^{2-5} = x^{-3} = \frac{1}{x^3}$ . The negative exponent tells you the result goes in the denominator.

Does this rule work with any base?

Absolutely! The quotient rule $\frac{a^m}{a^n} = a^{m-n}$ works for any base (numbers, variables, or expressions) as long as the base is exactly the same in both numerator and denominator.

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