Simplify the Exponential Expression: (3×7)^(2x+5) ÷ (3×7)^(x+3)

Q: Why do we subtract exponents when dividing?

Think of it as canceling out! When you divide $ a^5 ÷ a^3 $, you're really canceling 3 of the 5 factors, leaving $ a^2 $. The math shortcut is subtract: 5 - 3 = 2.

Q: Does it matter that the base is (3×7) instead of a single number?

Not at all! The quotient rule works for any base, whether it's a single number, a product like (3×7), or even a variable. The rule $ \frac{a^m}{a^n} = a^{m-n} $ applies to any base.

Q: What if I get a negative exponent?

Negative exponents are perfectly valid! $ a^{-n} = \frac{1}{a^n} $. In this problem, if x were smaller, you might get a negative result, which just means the answer would be a fraction.

Q: Can I simplify (3×7) to 21 first?

You can, but it's not necessary! The problem keeps it as (3×7), so your final answer should too. Both $ (3×7)^{x+2} $ and $ 21^{x+2} $ are correct.

Q: How do I subtract (x+3) from (2x+5)?

Be careful with the parentheses! $ (2x+5) - (x+3) = 2x + 5 - x - 3 $. The negative sign applies to both terms inside the second parentheses: x becomes -x and 3 becomes -3.

Quotient Rule with Same Base Expressions

Insert the corresponding expression:

$\frac{\left(3\times7\right)^{2x+5}}{\left(3\times7\right)^{x+3}}=$

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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 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

00:24 Let's open parentheses properly

00:27 Negative times positive always equals negative

00:35 Let's group terms

00:37 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:

$\frac{\left(3\times7\right)^{2x+5}}{\left(3\times7\right)^{x+3}}=$

Step-by-step solution

To solve the problem $\frac{\left(3\times7\right)^{2x+5}}{\left(3\times7\right)^{x+3}}=$ , we need to apply the Power of a Quotient Rule for Exponents.

The Power of a Quotient Rule states that $\frac{a^m}{a^n} = a^{m-n}$ where $a$ is a nonzero number and $m$ and $n$ are integers. In this expression, $a$ will be equal to $(3 \times 7)$ .

Start by writing the expression in a simplified form using the rule:

The numerator is $\left(3 \times 7\right)^{2x + 5}$
The denominator is $\left(3 \times 7\right)^{x + 3}$

Applying the quotient rule:

$\frac{\left(3 \times 7\right)^{2x+5}}{\left(3 \times 7\right)^{x+3}} = \left(3 \times 7\right)^{(2x+5) - (x+3)}$

Now we simplify the exponent:

$(2x + 5) - (x + 3) = 2x + 5 - x - 3$
Combine like terms: $2x - x + 5 - 3 = x + 2$

Thus, $\frac{\left(3 \times 7\right)^{2x+5}}{\left(3 \times 7\right)^{x+3}} = \left(3 \times 7\right)^{x+2}$ .

The solution to the question is: $\left(3 \times 7\right)^{x+2}$

Final Answer

$\left(3\times7\right)^{x+2}$

Key Points to Remember

Essential concepts to master this topic

Quotient Rule: When dividing same bases, subtract the exponents
Technique: $a^m ÷ a^n = a^{m-n}$ becomes $(3×7)^{(2x+5)-(x+3)}$
Check: Substitute a value: when x=1, $(3×7)^3 ÷ (3×7)^4 = (3×7)^{-1}$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting
Don't add the exponents (2x+5) + (x+3) = 3x+8! This gives the multiplication rule instead of division. Division means you subtract exponents, so always use (2x+5) - (x+3) = x+2.

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 $a^5 ÷ a^3$ , you're really canceling 3 of the 5 factors, leaving $a^2$ . The math shortcut is subtract: 5 - 3 = 2.

Does it matter that the base is (3×7) instead of a single number?

Not at all! The quotient rule works for any base, whether it's a single number, a product like (3×7), or even a variable. The rule $\frac{a^m}{a^n} = a^{m-n}$ applies to any base.

What if I get a negative exponent?

Negative exponents are perfectly valid! $a^{-n} = \frac{1}{a^n}$ . In this problem, if x were smaller, you might get a negative result, which just means the answer would be a fraction.

Can I simplify (3×7) to 21 first?

You can, but it's not necessary! The problem keeps it as (3×7), so your final answer should too. Both $(3×7)^{x+2}$ and $21^{x+2}$ are correct.

How do I subtract (x+3) from (2x+5)?

Be careful with the parentheses! $(2x+5) - (x+3) = 2x + 5 - x - 3$ . The negative sign applies to both terms inside the second parentheses: x becomes -x and 3 becomes -3.

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