Simplify the Expression: 6³ × 6⁻⁴ × 6⁷ Using Exponent Rules

Q: Why do we add exponents instead of multiplying them?

The multiplication rule for exponents states: $ a^m \times a^n = a^{m+n} $. This is because you're combining groups of the same base. Think of it as 6 × 6 × 6 combined with 1/(6 × 6 × 6 × 6) combined with 6 × 6 × 6 × 6 × 6 × 6 × 6.

Q: How do I handle the negative exponent -4?

A negative exponent means reciprocal: $ 6^{-4} = \frac{1}{6^4} $. When adding exponents, treat it like regular algebra: 3 + (-4) + 7. The negative sign stays with the 4.

Q: Can I work from left to right instead?

Yes! You can solve $ 6^3 \times 6^{-4} = 6^{3-4} = 6^{-1} $ first, then $ 6^{-1} \times 6^7 = 6^{-1+7} = 6^6 $. The final answer is the same!

Q: What's the difference between 6⁻⁴ and -6⁴?

Big difference! $ 6^{-4} = \frac{1}{6^4} $ (positive fraction), while $ -6^4 = -(6^4) $ (negative number). The negative sign's position matters!

Q: How can I check my final answer of 6⁶?

Calculate the actual value: $ 6^6 = 46656 $. You can also verify by computing each part separately and multiplying: $ 216 \times \frac{1}{1296} \times 279936 = 46656 $ ✓

Exponent Rules with Mixed Signs

Reduce the following equation:

$6^3\times6^{-4}\times6^7=$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Reduce the following equation:

$6^3\times6^{-4}\times6^7=$

Step-by-step solution

To solve the expression $6^3 \times 6^{-4} \times 6^7$ , we need to apply the rules of exponents, specifically the multiplication of powers. When we multiply powers with the same base, we add their exponents.

First, let's identify the base and the exponents in the expression:

The base is 6.
The exponents are 3, -4, and 7.

Using the exponent multiplication rule, we sum the exponents:

$6^3 \times 6^{-4} \times 6^7= 6^{3 + (-4) + 7}$

So, the solution is:

$6^{3-4+7}$

Final Answer

$6^{3-4+7}$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying powers with same base, add the exponents
Technique: Combine exponents: 3 + (-4) + 7 = 6
Check: Verify by expanding: 6³ = 216, 6⁻⁴ = 1/1296, 6⁷ = 279936 ✓

Common Mistakes

Avoid these frequent errors

Ignoring negative exponent signs when adding
Don't add 3 + 4 + 7 = 14 when you see 6³ × 6⁻⁴ × 6⁷! This ignores the negative sign and gives $6^{14}$ instead of $6^6$ . Always pay attention to negative signs: 3 + (-4) + 7 = 6.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do we add exponents instead of multiplying them?

The multiplication rule for exponents states: $a^m \times a^n = a^{m+n}$ . This is because you're combining groups of the same base. Think of it as 6 × 6 × 6 combined with 1/(6 × 6 × 6 × 6) combined with 6 × 6 × 6 × 6 × 6 × 6 × 6.

How do I handle the negative exponent -4?

A negative exponent means reciprocal: $6^{-4} = \frac{1}{6^4}$ . When adding exponents, treat it like regular algebra: 3 + (-4) + 7. The negative sign stays with the 4.

Can I work from left to right instead?

Yes! You can solve $6^3 \times 6^{-4} = 6^{3-4} = 6^{-1}$ first, then $6^{-1} \times 6^7 = 6^{-1+7} = 6^6$ . The final answer is the same!

What's the difference between 6⁻⁴ and -6⁴?

Big difference! $6^{-4} = \frac{1}{6^4}$ (positive fraction), while $-6^4 = -(6^4)$ (negative number). The negative sign's position matters!

How can I check my final answer of 6⁶?

Calculate the actual value: $6^6 = 46656$ . You can also verify by computing each part separately and multiplying: $216 \times \frac{1}{1296} \times 279936 = 46656$ ✓

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