Compare (5×3)² and Reciprocal of (4×4)⁻²: Insert > < or =

Q: Why does $ \frac{1}{16^{-2}} $ become 256 instead of a tiny fraction?

Great question! Since $ 16^{-2} = \frac{1}{16^2} = \frac{1}{256} $, taking the reciprocal gives us $ \frac{1}{\frac{1}{256}} = 256 $. The reciprocal of a reciprocal brings you back to the original!

Q: How do I remember the negative exponent rule?

Think of it as "flip and drop the negative": $ 5^{-3} $ becomes $ \frac{1}{5^3} $. The negative sign means "put it in the denominator" and make the exponent positive.

Q: Should I always convert negative exponents first?

Yes! It's much easier to work with positive exponents. Convert $ a^{-n} $ to $ \frac{1}{a^n} $ early in your solution to avoid confusion later.

Q: What if I calculated (5×3)² as 5²×3² = 25×9 = 225?

Perfect! That's absolutely correct. You can either calculate $ (5×3)^2 = 15^2 = 225 $ or use the power rule: $ (5×3)^2 = 5^2 × 3^2 = 25 × 9 = 225 $. Both methods work!

Q: How do I compare 225 and 256 quickly?

Since both are perfect squares, you can think: $ 225 = 15^2 $ and $ 256 = 16^2 $. Since 15 , we know \( 15^2 , so 225

Exponent Rules with Negative Powers

Insert the compatible sign:

$>,<,=$

$(5\times3)^2\Box\frac{1}{\left(4\times4\right)^{-2}}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Choose the appropriate sign

00:06 First let's simplify the left side

00:15 Let's calculate the multiplication

00:19 Now let's simplify the right side

00:30 Let's calculate the multiplication

00:34 Let's use the negative exponent formula

00:38 Any number (A) with a negative exponent(-N)

00:42 Equals the reciprocal number (1\A) with the opposite exponent (N)

00:45 Let's use this formula in our exercise

00:48 Let's substitute the reciprocal number and the opposite exponent

00:56 And this is the simplification for the right side

01:11 Now let's compare the simplifications and see which one is larger

01:20 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 compatible sign:

$>,<,=$

$(5\times3)^2\Box\frac{1}{\left(4\times4\right)^{-2}}$

Step-by-step solution

To solve this problem, let's evaluate each expression separately and then compare them:

Step 1: Evaluate $(5 \times 3)^2$

Calculate $5 \times 3$ :

5 \times 3 = 15

Now, square 15:

15^2 = 225

Step 2: Evaluate $\frac{1}{(4 \times 4)^{-2}}$

Calculate $4 \times 4$ :

4 \times 4 = 16

Apply the negative exponent rule: $a^{-n} = \frac{1}{a^n}$ . Thus, $16^{-2} = \frac{1}{16^2}$ .

Calculate $16^2$ :

16^2 = 256

Then, take the reciprocal as indicated by the negative exponent:

\frac{1}{16^{-2}} = \frac{1}{\frac{1}{256}} = 256

Step 3: Compare the two results

We have $(5 \times 3)^2 = 225$ and $\frac{1}{(4 \times 4)^{-2}} = 256$ .
Compare these values: $225 \lt 256$ .

The correct sign to place between the expressions is $<$ . Thus, the solution to the problem is:

(5 \times 3)^2 \lt \frac{1}{(4 \times 4)^{-2}}

Therefore, the correct answer is $\boxed{\lt}$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Negative Exponent Rule: $a^{-n} = \frac{1}{a^n}$ transforms negative powers to fractions
Reciprocal Property: $\frac{1}{x^{-n}} = x^n$ , so $\frac{1}{16^{-2}} = 16^2 = 256$
Check Work: Calculate each side separately then compare: 225 < 256 ✓

Common Mistakes

Avoid these frequent errors

Confusing reciprocals with negative exponents
Don't treat $\frac{1}{(4×4)^{-2}}$ as $\frac{1}{256}$ = 0.004! This ignores that the negative exponent already creates a reciprocal. Always remember: $\frac{1}{a^{-n}} = a^n$ , so the reciprocal of a reciprocal gives you the original power.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why does $\frac{1}{16^{-2}}$ become 256 instead of a tiny fraction?

Great question! Since $16^{-2} = \frac{1}{16^2} = \frac{1}{256}$ , taking the reciprocal gives us $\frac{1}{\frac{1}{256}} = 256$ . The reciprocal of a reciprocal brings you back to the original!

How do I remember the negative exponent rule?

Think of it as "flip and drop the negative": $5^{-3}$ becomes $\frac{1}{5^3}$ . The negative sign means "put it in the denominator" and make the exponent positive.

Should I always convert negative exponents first?

Yes! It's much easier to work with positive exponents. Convert $a^{-n}$ to $\frac{1}{a^n}$ early in your solution to avoid confusion later.

What if I calculated (5×3)² as 5²×3² = 25×9 = 225?

Perfect! That's absolutely correct. You can either calculate $(5×3)^2 = 15^2 = 225$ or use the power rule: $(5×3)^2 = 5^2 × 3^2 = 25 × 9 = 225$ . Both methods work!

How do I compare 225 and 256 quickly?

Since both are perfect squares, you can think: $225 = 15^2$ and $256 = 16^2$ . Since 15 < 16, we know $15^2 < 16^2$ , so 225 < 256.

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Compare (5×3)² and Reciprocal of (4×4)⁻²: Insert > < or =

Exponent Rules with Negative Powers

❤️ 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 does $\frac{1}{16^{-2}}$ become 256 instead of a tiny fraction?

How do I remember the negative exponent rule?

Should I always convert negative exponents first?

What if I calculated (5×3)² as 5²×3² = 25×9 = 225?

How do I compare 225 and 256 quickly?

🌟 Unlock Your Math Potential

More Questions

Power of a Product

Compare Powers: (10×3)^7 and the Reciprocal of (15×2)^-7

Solve: Product of Powers 20¹¹ × 4¹¹ × 2¹¹ × 5¹¹ × 3¹¹

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Calculate (2×4×6×3)^5: Product and Power Expression

Compare (5×3)² and Reciprocal of (4×4)⁻²: Insert > < or =

Exponent Rules with Negative Powers

❤️ 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 does 116−2 \frac{1}{16^{-2}} 16−21​ become 256 instead of a tiny fraction?

How do I remember the negative exponent rule?

Should I always convert negative exponents first?

What if I calculated (5×3)² as 5²×3² = 25×9 = 225?

How do I compare 225 and 256 quickly?

🌟 Unlock Your Math Potential

More Questions

Power of a Product

Compare Powers: (10×3)^7 and the Reciprocal of (15×2)^-7

Solve: Product of Powers 20¹¹ × 4¹¹ × 2¹¹ × 5¹¹ × 3¹¹

Calculate (8×9×10×5×6×3×7)⁵: Complex Exponent Problem

Calculate (2×4×6×3)^5: Product and Power Expression

Why does $\frac{1}{16^{-2}}$ become 256 instead of a tiny fraction?