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

Q: Why do we subtract exponents when dividing?

Think of it this way: $ \frac{x^7}{x^2} = \frac{x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x}{x \cdot x} $. You can cancel out two x's from top and bottom, leaving 5 x's on top, which is $ x^5 $!

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

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

Q: Does this work with numbers too?

Absolutely! For example, $ \frac{2^6}{2^4} = 2^{6-4} = 2^2 = 4 $. The quotient rule works for any base, whether it's a variable or a number.

Q: What if I multiply exponents by mistake?

Multiplying exponents gives you $ x^{14} $, which is way too big! Remember: multiplication of same bases means add exponents, but division means subtract exponents.

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

Use this memory trick: "Same operation, opposite exponent rule". Multiplication (×) uses addition (+) of exponents. Division (÷) uses subtraction (−) of exponents!

Division of Powers with Same Base

Insert the corresponding expression:

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's get started!

00:08 Today, we'll learn how to divide powers with the same base.

00:12 If you have a number, A, to the power of N, divided by the same base, A, to the power of M,

00:18 It equals A to the power of M minus N. Take away the exponents.

00:23 We'll apply this formula to solve our practice problem.

00:27 And that's how we figure out the solution! Well done!

Step-by-step written solution

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Understand the problem

Insert the corresponding expression:

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

Step-by-step solution

To solve the expression $\frac{x^7}{x^2}$ , we need to apply the power of a quotient rule for exponents. This rule states that $\frac{a^m}{a^n} = a^{m-n}$ , where $a$ is a nonzero number, and $m$ and $n$ are integers. This rule allows us to subtract the exponent in the denominator from the exponent in the numerator, given that the bases are the same.

Here's a step-by-step breakdown of applying the formula:

Identify the base $x$ which is common in both numerator and denominator.
Look at the exponents: the numerator has an exponent of 7 and the denominator has an exponent of 2.
According to the power of a quotient rule, subtract the exponent in the denominator from the exponent in the numerator: $7 - 2$ .
Perform the subtraction: $7 - 2 = 5$ .
The resulting exponent of $x$ is 5.

Therefore, $\frac{x^7}{x^2}$ simplifies to $x^5$ .

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

Final Answer

$x^5$

Key Points to Remember

Essential concepts to master this topic

Quotient Rule: When dividing same bases, subtract the exponents
Technique: $\frac{x^7}{x^2} = x^{7-2} = x^5$
Check: Multiply result by denominator: $x^5 \cdot x^2 = x^7$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting
Don't add the exponents 7 + 2 = 9 to get $x^9$ ! This gives multiplication, not division. Always subtract the bottom exponent from the top 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 we subtract exponents when dividing?

Think of it this way: $\frac{x^7}{x^2} = \frac{x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x}{x \cdot x}$ . You can cancel out two x's from top and bottom, leaving 5 x's on top, which is $x^5$ !

What if the bottom exponent is bigger than the top?

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

Does this work with numbers too?

Absolutely! For example, $\frac{2^6}{2^4} = 2^{6-4} = 2^2 = 4$ . The quotient rule works for any base, whether it's a variable or a number.

What if I multiply exponents by mistake?

Multiplying exponents gives you $x^{14}$ , which is way too big! Remember: multiplication of same bases means add exponents, but division means subtract exponents.

How can I remember when to add or subtract exponents?

Use this memory trick: "Same operation, opposite exponent rule". Multiplication (×) uses addition (+) of exponents. Division (÷) uses subtraction (−) of exponents!

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