Simplify the Exponential Expression: a^7y ÷ a^5x Using Power Rules

Q: Why do we subtract exponents when dividing?

Division is the opposite of multiplication. Since we add exponents when multiplying powers, we subtract them when dividing. Think of it as: $ a^m \div a^n = a^{m-n} $.

Q: Can I simplify 7y - 5x further?

No! Since y and x are different variables, you cannot combine or simplify $ 7y - 5x $. Leave it as is unless given specific values for the variables.

Q: What happens if both exponents were the same?

If you had $ \frac{a^{7y}}{a^{7y}} $, you'd get $ a^{7y-7y} = a^0 = 1 $. Any number (except 0) raised to the power of 0 equals 1!

Q: Does the base 'a' change during division?

Never! The base always stays the same when using exponent rules. Only the exponents change according to the operation you're performing.

Exponential Division with Variable Exponents

Solve the following exercise

$\frac{a^{7y}}{a^{5x}}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 When dividing powers with equal bases

00:07 The power of the result equals the difference between the exponents

00:10 We'll apply this formula to our exercise, and subtract the exponents

00:14 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following exercise

$\frac{a^{7y}}{a^{5x}}$

Step-by-step solution

Let's consider that in the given problem there is a fraction in both the numerator and denominator with terms of identical bases. Hence we use the property of division between terms of identical bases in order to solve the exercise:

$\frac{c^m}{c^n}=c^{m-n}$ We apply the previously mentioned property to the problem:

$\frac{a^{7y}}{a^{5x}}=a^{7y-5x}$ Therefore, the correct answer is option A.

Final Answer

$a^{7y-5x}$

Key Points to Remember

Essential concepts to master this topic

Division Rule: When dividing powers with same base, subtract exponents
Technique: $\frac{a^{7y}}{a^{5x}} = a^{7y-5x}$ by applying quotient rule
Check: Verify base remains unchanged and exponent is difference ✓

Common Mistakes

Avoid these frequent errors

Adding or multiplying exponents instead of subtracting
Don't add exponents like 7y + 5x = 12xy or multiply them! This ignores the division operation and gives completely wrong results. Always subtract the denominator's exponent from the numerator's exponent when dividing powers.

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?

Division is the opposite of multiplication. Since we add exponents when multiplying powers, we subtract them when dividing. Think of it as: $a^m \div a^n = a^{m-n}$ .

What if the exponents have variables like 7y and 5x?

Treat variable exponents just like numbers! The rule still applies: subtract the bottom exponent from the top exponent. So $a^{7y-5x}$ is your final answer.

Can I simplify 7y - 5x further?

No! Since y and x are different variables, you cannot combine or simplify $7y - 5x$ . Leave it as is unless given specific values for the variables.

What happens if both exponents were the same?

If you had $\frac{a^{7y}}{a^{7y}}$ , you'd get $a^{7y-7y} = a^0 = 1$ . Any number (except 0) raised to the power of 0 equals 1!

Does the base 'a' change during division?

Never! The base always stays the same when using exponent rules. Only the exponents change according to the operation you're performing.

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