Simplify the Expression: y^7 ÷ y^2 Using Exponent Rules

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

You still subtract! For example, $ \frac{y^2}{y^7} = y^{2-7} = y^{-5} $. The negative exponent means one divided by that positive power.

Q: Can I use this rule with different variables?

No! This rule only works when the bases are exactly the same. You cannot simplify $ \frac{x^5}{y^3} $ using this rule because x and y are different.

Q: What happens if both exponents are the same?

Great question! $ \frac{y^5}{y^5} = y^{5-5} = y^0 = 1 $. Any non-zero number to the power of 0 equals 1, which makes sense because anything divided by itself equals 1!

Exponent Division with Same Base

Insert the corresponding expression:

$\frac{y^7}{y^2}=$

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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)

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

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

00:10 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:

$\frac{y^7}{y^2}=$

Step-by-step solution

The given expression is $\frac{y^7}{y^2}$ .

To simplify this expression, we apply the Power of a Quotient Rule for exponents. This rule states that when you divide two expressions with the same base, you subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$ .

In this case, the base $y$ is the same for both the numerator and the denominator.

The exponent in the numerator is 7.
The exponent in the denominator is 2.

Thus, following the rule, we subtract the exponent of the denominator from the exponent of the numerator:

$y^{7-2} = y^5$

The solution to the question is: $y^5$

Final Answer

$y^5$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing same bases, subtract exponents: $\frac{a^m}{a^n} = a^{m-n}$
Technique: For $\frac{y^7}{y^2}$ , calculate 7 - 2 = 5 to get $y^5$
Check: Expand to verify: $\frac{y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y}{y \cdot y} = y^5$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting
Don't add 7 + 2 = 9 to get $y^9$ ! Addition is for multiplication, not division. Always subtract the denominator exponent from the numerator exponent when dividing same bases.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I subtract exponents when dividing?

When you divide, you're canceling out common factors! Think of $\frac{y^7}{y^2}$ as $\frac{y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y}{y \cdot y}$ . Two y's cancel out, leaving 5 y's = $y^5$ .

What if the bottom exponent is bigger than the top?

You still subtract! For example, $\frac{y^2}{y^7} = y^{2-7} = y^{-5}$ . The negative exponent means one divided by that positive power.

Can I use this rule with different variables?

No! This rule only works when the bases are exactly the same. You cannot simplify $\frac{x^5}{y^3}$ using this rule because x and y are different.

What happens if both exponents are the same?

Great question! $\frac{y^5}{y^5} = y^{5-5} = y^0 = 1$ . Any non-zero number to the power of 0 equals 1, which makes sense because anything divided by itself equals 1!

Do I need to write out all the y's to solve this?

Not at all! Once you understand the pattern, just use the rule directly. $\frac{y^7}{y^2} = y^{7-2} = y^5$ . Writing them out helps you understand why the rule works.

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