Simplify the Exponential Expression: 9^x ÷ 9^y

Q: Why do we subtract exponents when dividing?

Think of it this way: $ \frac{9^x}{9^y} $ means 9 multiplied x times, divided by 9 multiplied y times. The y copies of 9 in the denominator cancel with y copies in the numerator, leaving (x-y) copies of 9!

Q: What if the exponent in the denominator is larger than the numerator?

You still subtract! For example, $ \frac{9^2}{9^5} = 9^{2-5} = 9^{-3} $. The negative exponent means one divided by that positive power: $ \frac{1}{9^3} $.

Q: Can I use this rule with different bases like 9^x ÷ 3^y?

No! This rule only works when the bases are exactly the same. For different bases, you need to use other methods or convert to the same base if possible.

Q: How is this different from multiplying exponentials?

When multiplying same bases, you add exponents: $ 9^x \times 9^y = 9^{x+y} $. When dividing same bases, you subtract exponents: $ \frac{9^x}{9^y} = 9^{x-y} $.

Q: What if one of the exponents is zero?

Remember that any number to the power of zero equals 1! So $ \frac{9^x}{9^0} = \frac{9^x}{1} = 9^{x-0} = 9^x $, and $ \frac{9^0}{9^y} = \frac{1}{9^y} = 9^{0-y} = 9^{-y} $.

Exponential Division with Same Base

Insert the corresponding expression:

$\frac{9^x}{9^y}=$

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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)

00:06 equals the number (A) to the power of the difference of exponents (M-N)

00:08 We'll use this formula in our exercise

00:09 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:

$\frac{9^x}{9^y}=$

Step-by-step solution

We start with the expression: $\frac{9^x}{9^y}$ .
We need to simplify this expression using the Power of a Quotient Rule for exponents, which states that $\frac{a^m}{a^n} = a^{m-n}$ . Here, the base $a$ must be the same in both the numerator and the denominator, and we subtract the exponent of the denominator from the exponent of the numerator.

Applying this rule to our expression, we identify $a = 9$ , $m = x$ , and $n = y$ . So we have:

$a = 9$
$m = x$
$n = y$

Using the Power of a Quotient Rule, we therefore rewrite the expression as:

$a^{m-n} = 9^{x-y}$

Hence, the simplified expression of $\frac{9^x}{9^y}$ is $9^{x-y}$ .

The solution to the question is: $9^{x-y}$

Final Answer

$9^{x-y}$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing same bases, subtract exponents: a^m ÷ a^n = a^(m-n)
Technique: For 9^x ÷ 9^y, keep base 9 and subtract: x - y
Check: Verify by expanding: 9^3 ÷ 9^2 = 9^(3-2) = 9^1 = 9 ✓

Common Mistakes

Avoid these frequent errors

Adding or multiplying exponents instead of subtracting
Don't add x + y or multiply x × y when dividing = completely wrong operation! 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

$(3\times4\times5)^4=$

FAQ

Everything you need to know about this question

Why do we subtract exponents when dividing?

Think of it this way: $\frac{9^x}{9^y}$ means 9 multiplied x times, divided by 9 multiplied y times. The y copies of 9 in the denominator cancel with y copies in the numerator, leaving (x-y) copies of 9!

What if the exponent in the denominator is larger than the numerator?

You still subtract! For example, $\frac{9^2}{9^5} = 9^{2-5} = 9^{-3}$ . The negative exponent means one divided by that positive power: $\frac{1}{9^3}$ .

Can I use this rule with different bases like 9^x ÷ 3^y?

No! This rule only works when the bases are exactly the same. For different bases, you need to use other methods or convert to the same base if possible.

How is this different from multiplying exponentials?

When multiplying same bases, you add exponents: $9^x \times 9^y = 9^{x+y}$ . When dividing same bases, you subtract exponents: $\frac{9^x}{9^y} = 9^{x-y}$ .

What if one of the exponents is zero?

Remember that any number to the power of zero equals 1! So $\frac{9^x}{9^0} = \frac{9^x}{1} = 9^{x-0} = 9^x$ , and $\frac{9^0}{9^y} = \frac{1}{9^y} = 9^{0-y} = 9^{-y}$ .

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