Simplify the Expression: y^9 ÷ y^3 Using Exponent Rules

Q: Why do we subtract exponents when dividing?

Think of it as canceling out repeated multiplication! $ y^9 = y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y $ and $ y^3 = y \cdot y \cdot y $. When dividing, three y's cancel out, leaving $ y^6 $.

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

You still subtract! For example, $ \frac{y^3}{y^9} = y^{3-9} = y^{-6} $. The negative exponent means one divided by that positive power.

Q: Does this rule work with different bases like x and y?

No! The bases must be the same. You can only use $ \frac{a^m}{a^n} = a^{m-n} $ when both the numerator and denominator have identical bases.

Q: How can I remember when to add vs subtract exponents?

Multiplication = Add exponents: $ x^2 \cdot x^3 = x^5 $Division = Subtract exponents: $ \frac{x^5}{x^2} = x^3 $Think: Division undoes multiplication!

Q: What happens if the exponent becomes zero?

Any non-zero number to the power of zero equals 1! So $ \frac{y^5}{y^5} = y^{5-5} = y^0 = 1 $. This makes sense because any number divided by itself equals 1.

Exponent Division with Same Bases

Insert the corresponding expression:

$\frac{y^9}{y^3}=$

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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 division of powers with the same base (A) and different exponents

00:07 equals the same base (A) raised to the difference of the exponents (M-N)

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

00:13 And we'll compare the numbers to the variables in the formula

00:29 We'll keep the base and subtract between the exponents

00:40 And this is the solution to the question

Step-by-step written solution

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Understand the problem

Insert the corresponding expression:

$\frac{y^9}{y^3}=$

Step-by-step solution

To solve the expression $\frac{y^9}{y^3}$ , we will apply the rules of exponents, specifically the power of division rule, which states that when you divide like bases, you subtract the exponents.

Here are the steps to arrive at the solution:

Step 1: Identify and write down the expression: $\frac{y^9}{y^3}$ .
Step 2: Apply the division rule of exponents, which is $\frac{a^m}{a^n} = a^{m-n}$ , for any non-zero base $a$ .
Step 3: Using the division rule, subtract the exponent in the denominator from the exponent in the numerator: $y^{9-3}$
Step 4: Calculate the exponent: $9 - 3 = 6$
Step 5: Write down the simplified expression: $y^6$

Therefore, the expression $\frac{y^9}{y^3}$ simplifies to $y^6$ .

Final Answer

$y^6$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing same bases, subtract the exponents
Technique: For $\frac{y^9}{y^3}$ , calculate 9 - 3 = 6
Check: Verify $y^6 \cdot y^3 = y^9$ by adding exponents ✓

Common Mistakes

Avoid these frequent errors

Adding exponents when dividing
Don't add 9 + 3 = 12 to get $y^{12}$ ! Adding exponents is for multiplication, not division. Always subtract the bottom exponent from the top: $\frac{y^9}{y^3} = y^{9-3} = y^6$ .

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! $y^9 = y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y$ and $y^3 = y \cdot y \cdot y$ . When dividing, three y's cancel out, leaving $y^6$ .

What if the bottom exponent is larger than the top?

You still subtract! For example, $\frac{y^3}{y^9} = y^{3-9} = y^{-6}$ . The negative exponent means one divided by that positive power.

Does this rule work with different bases like x and y?

No! The bases must be the same. You can only use $\frac{a^m}{a^n} = a^{m-n}$ when both the numerator and denominator have identical bases.

How can I remember when to add vs subtract exponents?

Multiplication = Add exponents: $x^2 \cdot x^3 = x^5$
Division = Subtract exponents: $\frac{x^5}{x^2} = x^3$
Think: Division undoes multiplication!

What happens if the exponent becomes zero?

Any non-zero number to the power of zero equals 1! So $\frac{y^5}{y^5} = y^{5-5} = y^0 = 1$ . This makes sense because any number divided by itself equals 1.

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