Simplify (5a)^10 ÷ (5a)^2: Power Division with Compound Base

Q: Why do we subtract exponents when dividing?

Think of it as canceling out repeated multiplication! $ \frac{(5a)^{10}}{(5a)^2} $ means we have 10 copies of (5a) on top and 2 copies on bottom. After canceling, we're left with 10 - 2 = 8 copies.

Q: What if the base has multiplication like (5×a)?

Treat the entire expression (5×a) as one single base. The rule $ \frac{b^m}{b^n} = b^{m-n} $ still applies perfectly, where b = (5×a).

Q: Can I simplify 5×a first before applying the rule?

Not necessary! The compound base (5×a) works perfectly as-is. Keep it together to avoid confusion and apply the division rule directly.

Q: What if I get confused between multiplication and division of exponents?

Remember: Multiply bases → ADD exponents. Divide bases → SUBTRACT exponents. Division means taking away, so we subtract!

Q: How can I double-check my answer?

Expand a small example: $ \frac{(5a)^3}{(5a)^1} = \frac{(5a)(5a)(5a)}{(5a)} = (5a)^2 $. Notice how 3 - 1 = 2 matches!

Exponent Division with Same Base

Insert the corresponding expression:

$\frac{\left(5\times a\right)^{10}}{\left(5\times a\right)^2}=$

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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 the number (A) to the power of the difference of exponents (M-N)

00:10 We'll use this formula in our exercise

00:15 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:

$\frac{\left(5\times a\right)^{10}}{\left(5\times a\right)^2}=$

Step-by-step solution

To solve the given expression $\frac{\left(5\times a\right)^{10}}{\left(5\times a\right)^2}$ , we apply the Power of a Quotient Rule for exponents. This rule states that $\frac{b^m}{b^n} = b^{m-n}$ , where 'b' is a base and 'm' and 'n' are exponents.

In the given expression, the base is $5 \times a$ , and the exponents are 10 and 2, respectively.

Following the rule, we subtract the exponent in the denominator from the exponent in the numerator:

Numerator exponent: 10
Denominator exponent: 2
Subtract the exponents: $10 - 2 = 8$

Thus, we rewrite the expression as

\left(5\times a\right)^8

The solution to the question is:

\left(5\times a\right)^8

Final Answer

$\left(5\times a\right)^8$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing same bases, subtract exponents: $\frac{b^m}{b^n} = b^{m-n}$
Technique: Identify base (5a), subtract exponents: 10 - 2 = 8
Check: Expand both forms to verify $(5a)^{10} ÷ (5a)^2 = (5a)^8$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting
Don't add 10 + 2 = 12 to get $(5a)^{12}$ ! Addition is for multiplication, not division. This creates a much larger result than correct. Always subtract the bottom exponent from the top exponent when dividing.

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 as canceling out repeated multiplication! $\frac{(5a)^{10}}{(5a)^2}$ means we have 10 copies of (5a) on top and 2 copies on bottom. After canceling, we're left with 10 - 2 = 8 copies.

What if the base has multiplication like (5×a)?

Treat the entire expression (5×a) as one single base. The rule $\frac{b^m}{b^n} = b^{m-n}$ still applies perfectly, where b = (5×a).

Can I simplify 5×a first before applying the rule?

Not necessary! The compound base (5×a) works perfectly as-is. Keep it together to avoid confusion and apply the division rule directly.

What if I get confused between multiplication and division of exponents?

Remember: Multiply bases → ADD exponents. Divide bases → SUBTRACT exponents. Division means taking away, so we subtract!

How can I double-check my answer?

Expand a small example: $\frac{(5a)^3}{(5a)^1} = \frac{(5a)(5a)(5a)}{(5a)} = (5a)^2$ . Notice how 3 - 1 = 2 matches!

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