Reduce the Expression: 4³ × 4⁻⁵ Using Laws of Exponents

Q: Why do I add the exponents instead of multiplying them?

The product rule for exponents says $ a^m \times a^n = a^{m+n} $. Think of it this way: $ 4^3 $ means 4×4×4, and $ 4^{-5} $ means $ \frac{1}{4^5} $, so you're combining these powers by adding their exponents.

Q: What does a negative exponent like -2 actually mean?

A negative exponent means "one over the positive power". So $ 4^{-2} = \frac{1}{4^2} = \frac{1}{16} $. It's not a negative number - it's a positive fraction!

Q: How do I handle adding a positive and negative exponent?

Just use regular addition rules! $ 3 + (-5) = 3 - 5 = -2 $. Remember that adding a negative number is the same as subtracting that number.

Q: Can I simplify this further than 4^-2?

$ 4^{-2} $ is already in its simplest exponential form. You could write it as $ \frac{1}{16} $, but the exponential form is usually preferred unless specifically asked for the decimal value.

Q: What if the bases were different numbers?

The product rule only works with the same base. If you had $ 4^3 \times 3^{-5} $, you couldn't combine the exponents because the bases (4 and 3) are different. You'd need to calculate each power separately first.

Exponent Laws with Negative Powers

Reduce the following equation:

$4^3\times4^{-5}=$

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Understand the problem

Reduce the following equation:

$4^3\times4^{-5}=$

Step-by-step solution

To solve the expression $4^3 \times 4^{-5}$ , we need to apply the multiplication of powers rule. This rule states that when you multiply two powers with the same base, you can add their exponents. Mathematically, this is expressed as:

$a^m \times a^n = a^{m+n}$

In our case, the base $a$ is 4, and the exponents $m$ and $n$ are 3 and -5, respectively.

Applying the rule:

$4^3 \times 4^{-5} = 4^{3 + (-5)}$

Simplifying the exponent:

$3 + (-5) = -2$

So, the expression simplifies to:

$4^{-2}$

This is the reduced form of the given equation.

Final Answer

$4^{-2}$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying same bases, add the exponents together
Technique: $4^3 \times 4^{-5} = 4^{3+(-5)} = 4^{-2}$
Check: Verify by calculating: $4^3 = 64$ , $4^{-5} = \frac{1}{1024}$ , multiply to get $\frac{1}{16} = 4^{-2}$ ✓

Common Mistakes

Avoid these frequent errors

Multiplying the exponents instead of adding them
Don't multiply 3 × (-5) = -15 to get $4^{-15}$ ! This gives a completely wrong answer because you're confusing the power rule with the product rule. Always add exponents when multiplying same bases: $a^m \times a^n = a^{m+n}$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I add the exponents instead of multiplying them?

The product rule for exponents says $a^m \times a^n = a^{m+n}$ . Think of it this way: $4^3$ means 4×4×4, and $4^{-5}$ means $\frac{1}{4^5}$ , so you're combining these powers by adding their exponents.

What does a negative exponent like -2 actually mean?

A negative exponent means "one over the positive power". So $4^{-2} = \frac{1}{4^2} = \frac{1}{16}$ . It's not a negative number - it's a positive fraction!

How do I handle adding a positive and negative exponent?

Just use regular addition rules! $3 + (-5) = 3 - 5 = -2$ . Remember that adding a negative number is the same as subtracting that number.

Can I simplify this further than 4^-2?

$4^{-2}$ is already in its simplest exponential form. You could write it as $\frac{1}{16}$ , but the exponential form is usually preferred unless specifically asked for the decimal value.

What if the bases were different numbers?

The product rule only works with the same base. If you had $4^3 \times 3^{-5}$ , you couldn't combine the exponents because the bases (4 and 3) are different. You'd need to calculate each power separately first.

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