Expand b^7: Calculating the Seventh Power Expression

Exponent Expansion with Power Decomposition

Expand the following expression:

b7= b^7=

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:05 Let's find which expressions match the original one.
00:09 Remember, when we multiply with the same base, A, we add their exponents.
00:15 So, A raised to N plus M equals this new exponent.
00:20 Let's use this rule for our problem now.
00:23 Keep the base the same and add the exponents.
00:27 Notice, this expression isn't like the original one.
00:31 We'll apply the same rule to simplify the other expressions.
00:36 Look, this expression does equal the original!
00:39 And here, this one doesn't match the original.
00:45 Here the exponents are just adding, so no special rule is used.
00:49 And that gives us our solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Expand the following expression:

b7= b^7=

2

Step-by-step solution

To solve this problem, we'll follow these steps:

  • Step 1: Identify the power that needs expansion: b7 b^7 .
  • Step 2: Use the rule for the multiplication of powers to express this as a product of smaller powers of b b .
  • Step 3: Decompose 7 into a sum of smaller numbers and express the power as a product of smaller terms.

Now, let's apply these steps:

Step 1: We have b7 b^7 and need to write it as a product of powers.

Step 2: Recall the rule for multiplication of powers, ba×bb=ba+b b^a \times b^b = b^{a+b} . We will express 7 as the sum of smaller numbers.

Step 3: Choose smaller exponents that add up to 7. Here, we can use 1+2+4=7 1 + 2 + 4 = 7 . Therefore, b7=b1×b2×b4 b^7 = b^1 \times b^2 \times b^4 .

Therefore, the expanded form of the expression is b1×b2×b4 b^1 \times b^2 \times b^4 .

3

Final Answer

b1×b2×b4 b^1\times b^2\times b^4

Key Points to Remember

Essential concepts to master this topic
  • Power Rule: When multiplying same bases, add exponents together
  • Technique: Decompose 7 as 1+2+4 to get b1×b2×b4 b^1 \times b^2 \times b^4
  • Check: Verify 1+2+4=7, so the expanded form equals original b7 b^7 ✓

Common Mistakes

Avoid these frequent errors
  • Using addition instead of multiplication for exponent expansion
    Don't write b1+b7 b^1 + b^7 when expanding = completely different expression! Addition changes the mathematical meaning entirely. Always use multiplication when expanding powers: b7=ba×bb×bc b^7 = b^a \times b^b \times b^c where a+b+c=7.

Practice Quiz

Test your knowledge with interactive questions

\( 112^0=\text{?} \)

FAQ

Everything you need to know about this question

Why can't I just write b1×b7 b^1 \times b^7 ?

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While b1×b7=b8 b^1 \times b^7 = b^8 mathematically, this doesn't expand b7 b^7 into smaller parts. Expansion means breaking it down into smaller exponents that add up to 7.

Are there other ways to expand b7 b^7 ?

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Yes! You could use b3×b4 b^3 \times b^4 or b2×b5 b^2 \times b^5 or even b1×b1×b5 b^1 \times b^1 \times b^5 . As long as the exponents add up to 7, it's correct!

How do I know which exponents add up to 7?

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Think of it like making change! You need combinations that total 7. Try

  • 1+6=7
  • 2+5=7
  • 3+4=7
  • 1+2+4=7
Pick any combination that works for your problem.

What's the difference between b7 b^7 and 7b 7b ?

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b7 b^7 means b multiplied by itself 7 times, while 7b 7b means 7 multiplied by b once. These are completely different! b7=b×b×b×b×b×b×b b^7 = b \times b \times b \times b \times b \times b \times b

Why is 1+2+3=6 1+2+3=6 wrong for this problem?

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Because we need exponents that add up to 7, not 6! If you use b1×b2×b3=b6 b^1 \times b^2 \times b^3 = b^6 , you get the wrong power entirely. Always check your sum matches the target exponent.

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