Simplify the Expression: a^(7+x)/a^(10-2x) Using Exponent Rules

Q: Why do I subtract the exponents instead of dividing them?

The division rule for exponents says $ \frac{a^m}{a^n} = a^{m-n} $. Think of it as canceling: $ a^7 = a \times a \times a \times a \times a \times a \times a $, so when you divide, you're removing common factors!

Q: What happens with the negative sign in -(10-2x)?

The negative sign means multiply everything inside by -1. So -(10-2x) becomes -10+2x. Don't forget to change the sign of every term inside the parentheses!

Q: Can I simplify a^{-3+3x} further?

This is already in simplest form! You could factor out the 3: $ a^{3(x-1)} $, but $ a^{-3+3x} $ is the standard simplified answer.

Q: What if the base was different in numerator and denominator?

The division rule only works with the same base! If you had $ \frac{a^5}{b^3} $, you cannot combine them. The bases must be identical to use $ \frac{a^m}{a^n} = a^{m-n} $.

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 powers

00:11 We'll apply this formula to our exercise, and subtract the powers

00:22 Let's calculate the power

00:35 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Complete the following exercise:

$\frac{a^{7+x}}{a^{10-2x}}$

2

Step-by-step solution

In this problem there is a fraction where both the numerator and denominator contain terms with identical bases. Therefore we will apply the division law between terms with identical bases in order to solve the exercise:

$\frac{c^m}{c^n}=c^{m-n}$

Let's apply the aforementioned law:

$\frac{a^{7+x}}{a^{10-2x}}=a^{7+x-(10-2x)}$

In the first stage, we used the above exponents law and then proceeded to perform a simple subtraction between the exponent of the term in the numerator and the exponent of the term in the denominator, whilst using a common base. Since the term's exponent in the denominator is a two-term algebraic expression, we performed the subtraction carefully using parentheses.

Proceed to simplify the exponential expression, and apply the distribution law in order to expand the parentheses, whilst remembering that the (-) sign before the parentheses is actually a multiplication operation by minus 1:

$a^{7+x-(10-2x)}=a^{7+x-10+2x}=a^{-3+3x}$

In the second stage we combined like terms in the exponent.

In doing so we obtained the most simplified form of our expression.

Therefore, the correct answer is B.

3

Final Answer

$a^{-3+3x}$

Key Points to Remember

Essential concepts to master this topic

Division Rule: When dividing same bases, subtract exponents: $\frac{a^m}{a^n} = a^{m-n}$
Technique: Subtract carefully: $7+x-(10-2x) = 7+x-10+2x$
Check: Combine like terms to verify: 7-10=-3 and x+2x=3x gives $a^{-3+3x}$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to distribute the negative sign when subtracting exponents
Don't write 7+x-(10-2x) as 7+x-10-2x = wrong answer a^{-3-x}! The negative sign before parentheses means multiply by -1, so -2x becomes +2x. Always distribute: -(10-2x) = -10+2x.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I subtract the exponents instead of dividing them?

+

The division rule for exponents says $\frac{a^m}{a^n} = a^{m-n}$ . Think of it as canceling: $a^7 = a \times a \times a \times a \times a \times a \times a$ , so when you divide, you're removing common factors!

What happens with the negative sign in -(10-2x)?

+

The negative sign means multiply everything inside by -1. So -(10-2x) becomes -10+2x. Don't forget to change the sign of every term inside the parentheses!

How do I combine like terms in the exponent?

+

Group constants with constants and variables with variables: $7+x-10+2x = (7-10) + (x+2x) = -3+3x$ . Always combine like terms separately!

Can I simplify a^{-3+3x} further?

+

This is already in simplest form! You could factor out the 3: $a^{3(x-1)}$ , but $a^{-3+3x}$ is the standard simplified answer.

What if the base was different in numerator and denominator?

+

The division rule only works with the same base! If you had $\frac{a^5}{b^3}$ , you cannot combine them. The bases must be identical to use $\frac{a^m}{a^n} = a^{m-n}$ .

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Simplify the Expression: a^(7+x)/a^(10-2x) Using Exponent Rules

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