Simplify the Expression: (3×14)^(a+1) ÷ (3×14)^2

Q: Why do we subtract exponents when dividing?

Think of it as canceling out! When you divide $ x^5 ÷ x^2 $, you're canceling two x's from the bottom with two from the top, leaving $ x^{5-2} = x^3 $.

Q: What if the exponents have variables like (a+1)?

Treat them exactly the same way! Just subtract: (a+1) - 2 = a+1-2 = a-1. The quotient rule works with any exponents, even algebraic expressions.

Q: Do I need to calculate 3×14 first?

No! Keep the base as (3×14) throughout your work. The quotient rule applies to any base, whether it's a number, variable, or expression like (3×14).

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

Use this memory trick: Multiplication = Add exponents, Division = Subtract exponents. Same bases multiplying? Add! Same bases dividing? Subtract!

Q: What happens if the answer has a negative exponent?

That's perfectly fine! A negative exponent like $ x^{-2} $ just means $ \frac{1}{x^2} $. Your simplified form $ (3×14)^{a-1} $ could be negative if a

Exponent Rules with Quotient Division

Insert the corresponding expression:

$\frac{\left(3\times14\right)^{a+1}}{\left(3\times14\right)^2}=$

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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)

00:07 equals the number (A) to the power of the difference of exponents (M-N)

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

00:14 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:

$\frac{\left(3\times14\right)^{a+1}}{\left(3\times14\right)^2}=$

Step-by-step solution

We're given the expression:

$\frac{\left(3\times14\right)^{a+1}}{\left(3\times14\right)^2}$

The problem requires us to simplify this expression using the power of a quotient rule for exponents. This rule states that:

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

In our case, we identify:

$x = 3\times14$
$m = a+1$
$n = 2$

Applying the power of a quotient rule, we get:

$\frac{\left(3\times14\right)^{a+1}}{\left(3\times14\right)^2} = \left(3\times14\right)^{a+1-2}$

Therefore, the solution to the question is:

$\left(3\times14\right)^{a-1}$

Final Answer

$\left(3\times14\right)^{a+1-2}$

Key Points to Remember

Essential concepts to master this topic

Quotient Rule: When dividing same bases, subtract exponents: $\frac{x^m}{x^n} = x^{m-n}$
Technique: Identify the base (3×14) and exponents (a+1 and 2), then subtract
Check: Result should have same base with exponent (a+1-2) = (a-1) ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting
Don't add exponents like (a+1) + 2 = 2a+2 when dividing! This gives completely wrong results. Division means you subtract the bottom exponent from the top exponent. Always use the quotient rule: subtract exponents 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 as canceling out! When you divide $x^5 ÷ x^2$ , you're canceling two x's from the bottom with two from the top, leaving $x^{5-2} = x^3$ .

What if the exponents have variables like (a+1)?

Treat them exactly the same way! Just subtract: (a+1) - 2 = a+1-2 = a-1. The quotient rule works with any exponents, even algebraic expressions.

Do I need to calculate 3×14 first?

No! Keep the base as (3×14) throughout your work. The quotient rule applies to any base, whether it's a number, variable, or expression like (3×14).

How can I remember when to add vs subtract exponents?

Use this memory trick: Multiplication = Add exponents, Division = Subtract exponents. Same bases multiplying? Add! Same bases dividing? Subtract!

What happens if the answer has a negative exponent?

That's perfectly fine! A negative exponent like $x^{-2}$ just means $\frac{1}{x^2}$ . Your simplified form $(3×14)^{a-1}$ could be negative if a < 1.

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