Expand the Expression: Converting 4^-6 to Standard Form

Negative Exponents with Product Form Expansion

Expand the following expression:

4−6= 4^{-6}=

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Identify which expressions are equal to the original expression
00:03 According to the laws of exponents, the multiplication of powers with the same base (A)
00:06 equals the same base raised to the sum of the exponents (N+M)
00:09 We will apply this formula to our exercise
00:12 We'll maintain the base and add the exponents together
00:17 We can observe that this expression is not equal to the original expression
00:20 We'll use the same method in order to simplify the remaining expressions
00:26 This expression is equal to the original expression
00:38 This expression is not equal to the original expression
00:47 This expression is also not equal to the original expression
00:51 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Expand the following expression:

4−6= 4^{-6}=

2

Step-by-step solution

The problem asks us to expand the expression 4−6 4^{-6} using the rules of exponents.

To start, recognize that the negative exponent −6-6 can be split into smaller parts, which can be achieved by breaking it into two equal parts: −3+(−3) -3 + (-3) . This means we can rewrite 4−6 4^{-6} as:

4−6=4−3+(−3)=4−3×4−3 4^{-6} = 4^{-3 + (-3)} = 4^{-3} \times 4^{-3}

By expressing 4−6 4^{-6} as a product of two identical terms, 4−3×4−3 4^{-3} \times 4^{-3} , we have expanded the original expression correctly according to the rules of exponents. This uses the property of exponents that states am+n=am×an a^{m+n} = a^m \times a^n .

Thus, the expanded form of 4−6 4^{-6} is 4−3×4−3 4^{-3} \times 4^{-3} .

3

Final Answer

4−3×4−3 4^{-3}\times4^{-3}

Key Points to Remember

Essential concepts to master this topic
  • Product Rule: am+n=am×an a^{m+n} = a^m \times a^n applies to negative exponents
  • Technique: Split 4−6 4^{-6} as −6=−3+(−3) -6 = -3 + (-3) for equal factors
  • Check: Verify 4−3×4−3=4−3+(−3)=4−6 4^{-3} \times 4^{-3} = 4^{-3+(-3)} = 4^{-6} ✓

Common Mistakes

Avoid these frequent errors
  • Adding exponents instead of using multiplication
    Don't write 4−6=4−2+4−3 4^{-6} = 4^{-2} + 4^{-3} or 4−3+4−3 4^{-3} + 4^{-3} = wrong operation! Addition doesn't follow exponent rules and gives incorrect expansions. Always use multiplication with the product rule: 4−6=4−3×4−3 4^{-6} = 4^{-3} \times 4^{-3} .

Practice Quiz

Test your knowledge with interactive questions

\( (3\times4\times5)^4= \)

FAQ

Everything you need to know about this question

Why can't I just add the terms like 4−3+4−3 4^{-3} + 4^{-3} ?

+

Because exponent rules work with multiplication, not addition! When you have am×an a^m \times a^n , you add the exponents. But 4−3+4−3 4^{-3} + 4^{-3} equals 2×4−3 2 \times 4^{-3} , which is completely different from 4−6 4^{-6} .

How do I know which way to split the negative exponent?

+

You can split −6 -6 many ways: −1+(−5) -1 + (-5) , −2+(−4) -2 + (-4) , or −3+(−3) -3 + (-3) . The question asks for expansion, so any valid split works. Choose the one that makes the most sense for the context!

What's the difference between expanding and simplifying?

+

Expanding means writing as a product of simpler terms (like 4−3×4−3 4^{-3} \times 4^{-3} ). Simplifying means calculating the final decimal value. This problem asks you to expand, not simplify to a decimal.

Can I use different bases when expanding?

+

No! When expanding 4−6 4^{-6} , you must keep the same base (4) in all terms. You can only change the exponents using the product rule: am+n=am×an a^{m+n} = a^m \times a^n .

Why is 4−1×4−6 4^{-1} \times 4^{-6} wrong?

+

Because 4−1×4−6=4−1+(−6)=4−7 4^{-1} \times 4^{-6} = 4^{-1+(-6)} = 4^{-7} , not 4−6 4^{-6} ! The exponents must add up to the original exponent. Only −3+(−3)=−6 -3 + (-3) = -6 works correctly.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Exponents Rules questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime

More Questions

Click on any question to see the complete solution with step-by-step explanations

Multiplication of Powers

Evaluate the Negative Exponent Expression: 9^-7
Click to solve
Expand b^7: Calculating the Seventh Power Expression
Click to solve