Solve for the Base: Finding x in x^7 = (1/5)^7

Q: Why isn't the answer 5 since I see the number 5?

Great question! The key is to look at what's being multiplied repeatedly. We're multiplying $ \frac{1}{5} $ seven times, not 5. So our base is $ \frac{1}{5} $.

Q: How do I know when bases are equal in an equation?

When you have $ a^n = b^n $ and the exponents are the same, then a must equal b. This is because raising different numbers to the same power gives different results.

Q: What if the fraction was written differently?

No matter how it's written! Whether you see $ \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5}... $ or $ \left(\frac{1}{5}\right)^7 $, they mean the same thing - the base is $ \frac{1}{5} $.

Q: How can I double-check my answer?

Substitute your answer back! If you think the base is $ \frac{1}{5} $, calculate $ \left(\frac{1}{5}\right)^7 $ and see if it equals the right side of the original equation.

Q: Do I need to calculate the actual value of the seventh power?

Not necessarily! Since both sides have the same exponent (7), you just need to match the bases. The calculation $ \left(\frac{1}{5}\right)^7 $ would be very small, but recognizing the pattern is what matters.

Exponent Equations with Fractional Bases

Fill in the missing number:

$☐^7=\frac{1}{5}\cdot\frac{1}{5}\cdot\frac{1}{5}\cdot\frac{1}{5}\cdot\frac{1}{5}\cdot\frac{1}{5}\cdot\frac{1}{5}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Complete the missing number

00:03 Let's use the power formula

00:06 Any number (X) to the power of (N)

00:09 equals X multiplied by itself N times

00:16 Let's use this formula in our exercise

00:23 X is the number being multiplied

00:30 Let's count the number of multiplications to find the unknown N

00:36 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Fill in the missing number:

$☐^7=\frac{1}{5}\cdot\frac{1}{5}\cdot\frac{1}{5}\cdot\frac{1}{5}\cdot\frac{1}{5}\cdot\frac{1}{5}\cdot\frac{1}{5}$

Step-by-step solution

To solve this problem, we'll begin by simplifying the expression on the right side of the equation:

$☐^7 = \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5}$

Using the laws of exponents, multiplying the fraction $\frac{1}{5}$ by itself seven times can be expressed as:

$\left(\frac{1}{5}\right)^7$

Now, the equation becomes:

$☐^7 = \left(\frac{1}{5}\right)^7$

Since the exponents on both sides of the equation are the same, the bases must be equal as well. Therefore, $☐ = \frac{1}{5}$ .

Thus, the missing number is:

$\frac{1}{5}$

Final Answer

$\frac{1}{5}$

Key Points to Remember

Essential concepts to master this topic

Rule: When bases are equal, exponents equal means bases equal
Technique: Rewrite repeated multiplication $\frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5}$ as $\left(\frac{1}{5}\right)^7$
Check: Verify that $\left(\frac{1}{5}\right)^7 = \left(\frac{1}{5}\right)^7$ produces true statement ✓

Common Mistakes

Avoid these frequent errors

Assuming the answer is the denominator 5
Don't think the answer is 5 just because you see $\frac{1}{5}$ in the problem = wrong base! The repeated fraction $\frac{1}{5}$ means the base itself is $\frac{1}{5}$ , not 5. Always recognize that $\left(\frac{1}{5}\right)^7$ requires base $\frac{1}{5}$ .

Practice Quiz

Test your knowledge with interactive questions

$$ 11^2= $$

FAQ

Everything you need to know about this question

Why isn't the answer 5 since I see the number 5?

Great question! The key is to look at what's being multiplied repeatedly. We're multiplying $\frac{1}{5}$ seven times, not 5. So our base is $\frac{1}{5}$ .

How do I know when bases are equal in an equation?

When you have $a^n = b^n$ and the exponents are the same, then a must equal b. This is because raising different numbers to the same power gives different results.

What if the fraction was written differently?

No matter how it's written! Whether you see $\frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5}...$ or $\left(\frac{1}{5}\right)^7$ , they mean the same thing - the base is $\frac{1}{5}$ .

How can I double-check my answer?

Substitute your answer back! If you think the base is $\frac{1}{5}$ , calculate $\left(\frac{1}{5}\right)^7$ and see if it equals the right side of the original equation.

Do I need to calculate the actual value of the seventh power?

Not necessarily! Since both sides have the same exponent (7), you just need to match the bases. The calculation $\left(\frac{1}{5}\right)^7$ would be very small, but recognizing the pattern is what matters.

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