Simplify the Expression: 20^(x+y) ÷ 20^(a+y) Using Exponent Rules

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

The Quotient Rule for Exponents states that $ \frac{b^m}{b^n} = b^{m-n} $. This works because division is the opposite of multiplication, and when multiplying same bases, we add exponents.

Q: What happens to the y terms in this problem?

The y terms cancel out completely! When you subtract (a+y) from (x+y), you get x+y-a-y, and the +y and -y cancel each other, leaving just x-a.

Q: Do I need to worry about the base being 20?

No! The quotient rule works for any non-zero base. Whether it's 2, 10, 20, or any other number, you still just subtract the exponents when the bases are the same.

Q: How do I handle the parentheses in the exponent subtraction?

Always use parentheses around each complete exponent: (x+y)-(a+y). Then distribute the negative sign: x+y-a-y. This prevents sign errors!

Q: What if the variables were different in each exponent?

If there were no common terms to cancel (like if it was $ \frac{20^{x+y}}{20^{a+b}} $), your final answer would be $ 20^{x+y-a-b} $ with all four variable terms remaining.

Exponent Quotient Rules with Variable Terms

Insert the corresponding expression:

$\frac{20^{x+y}}{20^{a+y}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 Here's how it works.

00:11 Let's talk about dividing exponents.

00:14 If we have a number A raised to the power of N, divided by A raised to M,

00:19 it equals A to the power of M minus N.

00:24 Let's try using this formula in an exercise.

00:28 Remember to carefully expand the parentheses.

00:32 A negative number times a positive number is always negative.

00:36 Let's group similar terms together.

00:39 And that's how we find the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{20^{x+y}}{20^{a+y}}=$

Step-by-step solution

Let's start by analyzing the given expression $\frac{20^{x+y}}{20^{a+y}}$ .

We have a fraction with the same base number in both the numerator and the denominator.

According to the Power of a Quotient Rule for Exponents, for any non-zero number $b$ and integers $m$ and $n$ , $\frac{b^m}{b^n} = b^{m-n}$ .

This means we can subtract the exponents of the denominator from the exponents of the numerator. First, write down the exponents explicitly:

Numerator: $x + y$
Denominator: $a + y$

Next, apply the rule:

$\frac{20^{x+y}}{20^{a+y}} = 20^{(x+y)-(a+y)}$

Distribute the subtraction in the exponent:

$20^{x+y-a-y}$

Simplify the terms:

As the variable $y$ is present in both terms, they cancel each other out, resulting in:

$20^{x-a}$

This is the simplest form for the expression, and it matches the provided correct answer.

The solution to the question is: $20^{x-a}$

Final Answer

$20^{x-a}$

Key Points to Remember

Essential concepts to master this topic

Quotient Rule: When dividing same bases, subtract exponents: $\frac{b^m}{b^n} = b^{m-n}$
Technique: For $\frac{20^{x+y}}{20^{a+y}}$ , subtract: (x+y) - (a+y) = x+y-a-y
Check: Variable y cancels out leaving $20^{x-a}$ as the simplified form ✓

Common Mistakes

Avoid these frequent errors

Forgetting to use parentheses when subtracting exponents
Don't write x+y-a+y = x-a! Missing parentheses means you add y instead of subtracting it. This gives the wrong simplified form. Always use parentheses: (x+y)-(a+y) to properly distribute the subtraction.

Practice Quiz

Test your knowledge with interactive questions

$(3\times4\times5)^4=$

FAQ

Everything you need to know about this question

Why do we subtract the exponents instead of dividing them?

The Quotient Rule for Exponents states that $\frac{b^m}{b^n} = b^{m-n}$ . This works because division is the opposite of multiplication, and when multiplying same bases, we add exponents.

What happens to the y terms in this problem?

The y terms cancel out completely! When you subtract (a+y) from (x+y), you get x+y-a-y, and the +y and -y cancel each other, leaving just x-a.

Do I need to worry about the base being 20?

No! The quotient rule works for any non-zero base. Whether it's 2, 10, 20, or any other number, you still just subtract the exponents when the bases are the same.

How do I handle the parentheses in the exponent subtraction?

Always use parentheses around each complete exponent: (x+y)-(a+y). Then distribute the negative sign: x+y-a-y. This prevents sign errors!

What if the variables were different in each exponent?

If there were no common terms to cancel (like if it was $\frac{20^{x+y}}{20^{a+b}}$ ), your final answer would be $20^{x+y-a-b}$ with all four variable terms remaining.

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