Simplify (17×3)^17 ÷ (17×3)^11: Laws of Exponents Practice

Q: Why do we subtract exponents when dividing?

Think of it this way: $ \frac{a^5}{a^3} = \frac{a×a×a×a×a}{a×a×a} $. The three a's cancel out, leaving two a's, which is $ a^2 $. That's why 5 - 3 = 2!

Q: Can I simplify 17×3 to 51 first?

You could calculate $ 17×3 = 51 $ first, giving you $ \frac{51^{17}}{51^{11}} = 51^6 $. Both $ (17×3)^6 $ and $ 51^6 $ are correct, but leaving it as $ (17×3)^6 $ matches the original format.

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

If you have something like $ \frac{a^3}{a^7} $, you still subtract: 3 - 7 = -4, giving $ a^{-4} $. Negative exponents mean reciprocals!

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

Multiplication: $ a^m × a^n = a^{m+n} $ (ADD)Division: $ \frac{a^m}{a^n} = a^{m-n} $ (SUBTRACT)Think: More terms = add, fewer terms = subtract!

Quotient Rule with Same Bases

Insert the corresponding expression:

$\frac{\left(17\times3\right)^{17}}{\left(17\times3\right)^{11}}=$

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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

00:13 Let's calculate the power

00:15 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:

$\frac{\left(17\times3\right)^{17}}{\left(17\times3\right)^{11}}=$

Step-by-step solution

Let's start solving this equation step by step. The problem provided is:

$\frac{\left(17\times3\right)^{17}}{\left(17\times3\right)^{11}}=$

This problem involves the Power of a Quotient Rule for Exponents, which states:

If you have a quotient of terms with the same base, you can subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$ .

The terms in our problem already have the same base $(17\times3)$ . Therefore, we apply the rule directly:

$\frac{\left(17\times3\right)^{17}}{\left(17\times3\right)^{11}} = \left( 17 \times 3 \right)^{17-11}$

Simplifying the exponent gives

$17 - 11 = 6$

Thus, the expression simplifies to:

$\left(17 \times 3\right)^6$

Therefore, the solution to the question is:

$\left(17 \times 3\right)^6$

Final Answer

$\left(17\times3\right)^6$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing same bases, subtract the exponents
Technique: $\frac{a^m}{a^n} = a^{m-n}$ gives 17-11 = 6
Check: $(17×3)^6$ has base $(17×3)$ and exponent 6 ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting
Don't add the exponents 17 + 11 = 28! This gives $(17×3)^{28}$ which is completely wrong. Division of same bases requires subtracting exponents. Always remember: division means subtract, multiplication means add.

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: $\frac{a^5}{a^3} = \frac{a×a×a×a×a}{a×a×a}$ . The three a's cancel out, leaving two a's, which is $a^2$ . That's why 5 - 3 = 2!

What if the bases look different but are actually the same?

Great question! In this problem, both the numerator and denominator have the base $(17×3)$ . Even though it's written as a product, it's still one base, so we can apply the quotient rule directly.

Can I simplify 17×3 to 51 first?

You could calculate $17×3 = 51$ first, giving you $\frac{51^{17}}{51^{11}} = 51^6$ . Both $(17×3)^6$ and $51^6$ are correct, but leaving it as $(17×3)^6$ matches the original format.

What happens if the bottom exponent is bigger than the top?

If you have something like $\frac{a^3}{a^7}$ , you still subtract: 3 - 7 = -4, giving $a^{-4}$ . Negative exponents mean reciprocals!

How do I remember when to add vs subtract exponents?

Multiplication: $a^m × a^n = a^{m+n}$ (ADD)
Division: $\frac{a^m}{a^n} = a^{m-n}$ (SUBTRACT)

Think: More terms = add, fewer terms = subtract!

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Simplify (17×3)^17 ÷ (17×3)^11: Laws of Exponents Practice

Quotient Rule with Same Bases

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do we subtract exponents when dividing?

What if the bases look different but are actually the same?

Can I simplify 17×3 to 51 first?

What happens if the bottom exponent is bigger than the top?

How do I remember when to add vs subtract exponents?

🌟 Unlock Your Math Potential

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