Find the Missing Exponent: (1/5)^□ = (1/5) × (1/5) × (1/5) × (1/5) × (1/5)

Q: How do I know what number goes in the exponent?

Count how many times the base (the number being multiplied) appears in the multiplication. In this problem, $ \frac{1}{5} $ appears 5 times, so the exponent is 5.

Q: What's the difference between the base and the exponent?

The base is the number being multiplied repeatedly ($ \frac{1}{5} $ in this case). The exponent tells you how many times to multiply the base by itself.

Q: What if the base was a whole number like 3?

The same rule applies! If you see $ 3 \cdot 3 \cdot 3 \cdot 3 $, that equals $ 3^4 $ because 3 appears 4 times as a factor.

Q: How can I check if my answer is right?

Expand your exponential form back to multiplication. $ \left(\frac{1}{5}\right)^5 $ should give you exactly 5 copies of $ \frac{1}{5} $ multiplied together.

Exponent Rules with Repeated Multiplication

Fill in the missing number:

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:12 Let's find the missing number together.

00:15 We'll use the power formula.

00:18 It means any number, like X, raised to the power of N.

00:23 Is simply X multiplied by itself, N times. Got it?

00:30 Now, let's apply this in our exercise.

00:34 Remember, X is the number we're multiplying.

00:37 Let's count how many times, to find our unknown, N.

00:42 And there we have it! That's how we solve the problem.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Fill in the missing number:

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

Step-by-step solution

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

Step 1: Count the number of times $\frac{1}{5}$ is used as a factor in the product given.
Step 2: Use this count as the exponent in the expression $\frac{1}{5}^☐$ .

Now, let's work through each step:
Step 1: The expression $\frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5}$ shows that the base $\frac{1}{5}$ is used 5 times.
Step 2: Therefore, the exponent that makes the expression $\left( \frac{1}{5} \right)^☐$ equal to the product is 5.

Therefore, the missing number in the expression $\frac{1}{5}^☐$ is $5$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Definition: An exponent shows how many times to multiply the base
Counting: Count each factor: $\frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5}$ = 5 factors
Verify: Check that $\left(\frac{1}{5}\right)^5$ equals the original product ✓

Common Mistakes

Avoid these frequent errors

Confusing the base with the exponent
Don't use 1/5 as the exponent just because it appears in the problem = $\left(\frac{1}{5}\right)^{1/5}$ which is completely wrong! The exponent tells you HOW MANY times to multiply, not WHAT to multiply. Always count the number of identical factors being multiplied together.

Practice Quiz

Test your knowledge with interactive questions

$$ 11^2= $$

FAQ

Everything you need to know about this question

How do I know what number goes in the exponent?

Count how many times the base (the number being multiplied) appears in the multiplication. In this problem, $\frac{1}{5}$ appears 5 times, so the exponent is 5.

What's the difference between the base and the exponent?

The base is the number being multiplied repeatedly ( $\frac{1}{5}$ in this case). The exponent tells you how many times to multiply the base by itself.

Can I just multiply all the fractions together instead?

You could, but that's much harder! Using exponent rules is faster. Recognizing patterns like repeated multiplication helps you solve problems more efficiently.

What if the base was a whole number like 3?

The same rule applies! If you see $3 \cdot 3 \cdot 3 \cdot 3$ , that equals $3^4$ because 3 appears 4 times as a factor.

How can I check if my answer is right?

Expand your exponential form back to multiplication. $\left(\frac{1}{5}\right)^5$ should give you exactly 5 copies of $\frac{1}{5}$ multiplied together.

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