Solve the Equation: Finding the Base Number in x^7 = 1

Exponential Equations with Integer Bases

Fill in the missing number:

7=1 ☐^7=1

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:05 Let's find the missing number together.
00:08 We'll use the power formula.
00:11 So, any number X raised to the power of N, means X is multiplied by itself, N times.
00:19 Got it? Now, we'll use this formula in our exercise.
00:23 Let's break the power into single multiplications.
00:26 Remember, one raised to any number is always one.
00:34 And that's how we find the solution. Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Fill in the missing number:

7=1 ☐^7=1

2

Step-by-step solution

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

  • Step 1: Identify potential values for the missing number \square.
  • Step 2: Verify if it fits the equation 7=1☐^7 = 1.
  • Step 3: Consider alternative numbers from intuitive mathematics.

Now, let's work through each step:
Step 1: We hypothesize that the number could be 1, given that raising 1 to any power yields 1.
Step 2: Verify (1)7=1(1)^7 = 1, which is true since any real number 1 raised to any integer power remains 1.
Step 3: As a check for understanding: with odd powers greater than 1, attempts with 1-1 as choices lead to (1)7=1(-1)^7 = -1, which doesn’t meet the requirement, reinforcing =1☐ = 1.

Therefore, the solution to the problem is 1, choice number 4.

3

Final Answer

1

Key Points to Remember

Essential concepts to master this topic
  • Identity Property: Any number raised to any power equals itself only when that number is 1
  • Technique: Test 17=1×1×1×1×1×1×1=1 1^7 = 1 \times 1 \times 1 \times 1 \times 1 \times 1 \times 1 = 1
  • Check: Verify (1)7=1 (-1)^7 = -1 doesn't work since 11 -1 ≠ 1

Common Mistakes

Avoid these frequent errors
  • Thinking any number works for the equation
    Don't assume any number raised to the 7th power equals 1 = multiple wrong answers! Only 1 has this special property. Always test your answer: 27=128 2^7 = 128 , 37=2187 3^7 = 2187 , but 17=1 1^7 = 1 .

Practice Quiz

Test your knowledge with interactive questions

\( 0+0.2+0.6= \) ?

FAQ

Everything you need to know about this question

Why is 1 the only answer when there are 7th roots of unity?

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You're thinking of complex numbers! In real numbers, only 1 raised to any power gives 1. Complex 7th roots like e2πi/7 e^{2πi/7} aren't typically covered at this level.

What about negative numbers like -1?

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Great question! Let's check: (1)7=1 (-1)^7 = -1 because 7 is odd. Since 11 -1 ≠ 1 , this doesn't work. Odd powers of negative numbers stay negative!

Could 0 be an answer?

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07=0 0^7 = 0 , not 1! Zero raised to any positive power always equals zero, never 1.

How do I remember this rule about 1?

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Think of it this way: 1 is the multiplicative identity. When you multiply 1 by itself any number of times, you always get 1 back. It's like adding zeros - they don't change the result!

Are there other numbers that work for different powers?

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Yes! For x2=1 x^2 = 1 , both 1 and -1 work. For x4=1 x^4 = 1 , we have 1 and -1. The pattern depends on whether the exponent is even or odd.

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