Simplify the Expression: (1/5) Multiplied Four Times

Q: How do I count the factors correctly?

Count each $ \frac{1}{5} $ in the multiplication: $ \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} $. There are 4 factors, so the exponent is 4, giving us $ (\frac{1}{5})^4 $.

Q: Why can't I add the fractions instead?

We're multiplying the fractions, not adding them! When you see multiplication symbols (·), use exponent rules. Addition would give a completely different result.

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

The base is what's being repeated: $ \frac{1}{5} $. The exponent is how many times it appears: 4. So we write $ (\frac{1}{5})^4 $.

Q: Can I simplify this further?

You could calculate the numerical value: $ (\frac{1}{5})^4 = \frac{1}{625} $, but the question asks for the representation, so $ (\frac{1}{5})^4 $ is the correct form.

Q: What if the fractions were different?

If the fractions were different (like $ \frac{1}{2} \cdot \frac{1}{3} $), you couldn't use exponents. Exponent notation only works with identical factors.

Exponent Rules with Repeated Multiplication

Which of the following represents the expression below?

$\frac{1}{5}\cdot\frac{1}{5}\cdot\frac{1}{5}\cdot\frac{1}{5}$ ?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 First, let's convert to power.

00:12 Remember, when a number, M, is multiplied by itself, it's squared.

00:22 The exponent tells us how many times a number is multiplied by itself.

00:27 Now, let's count the repetitions in our question.

00:31 This gives us our exponent!

00:41 For a fraction, A divided by B, raised to N,

00:45 we raise both the numerator and denominator to the power of N.

00:50 Let's apply this rule to our exercise.

00:54 One to the power of any number, N, is always one.

00:58 And that's how we solve this problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Which of the following represents the expression below?

$\frac{1}{5}\cdot\frac{1}{5}\cdot\frac{1}{5}\cdot\frac{1}{5}$ ?

Step-by-step solution

To solve the problem, let's represent the repeated multiplication using exponents:

We start with the given expression:

$\frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5}$

Notice that $\frac{1}{5}$ is multiplied by itself four times. This can be expressed as a power:

$(\frac{1}{5})^4$

Hence, the correct representation of the given expression is $(\frac{1}{5})^4$ .

From the given choices, the correct option is Choice 4: $(\frac{1}{5})^4$ .

Therefore, the solution to the problem is $(\frac{1}{5})^4$ .

Final Answer

$(\frac{1}{5})^4$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying identical bases, add the exponents together
Technique: Count repetitions: $\frac{1}{5}$ appears 4 times = $(\frac{1}{5})^4$
Check: Verify by expanding the exponent back to multiplication ✓

Common Mistakes

Avoid these frequent errors

Counting repetitions incorrectly
Don't count 3 factors and write $(\frac{1}{5})^3$ = wrong representation! This happens when students miscount or confuse the number of multiplication signs with the number of factors. Always count each factor carefully: $\frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5}$ has 4 identical factors.

Practice Quiz

Test your knowledge with interactive questions

$$ 11^2= $$

FAQ

Everything you need to know about this question

How do I count the factors correctly?

Count each $\frac{1}{5}$ in the multiplication: $\frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5}$ . There are 4 factors, so the exponent is 4, giving us $(\frac{1}{5})^4$ .

Why can't I add the fractions instead?

We're multiplying the fractions, not adding them! When you see multiplication symbols (·), use exponent rules. Addition would give a completely different result.

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

The base is what's being repeated: $\frac{1}{5}$ . The exponent is how many times it appears: 4. So we write $(\frac{1}{5})^4$ .

Can I simplify this further?

You could calculate the numerical value: $(\frac{1}{5})^4 = \frac{1}{625}$ , but the question asks for the representation, so $(\frac{1}{5})^4$ is the correct form.

What if the fractions were different?

If the fractions were different (like $\frac{1}{2} \cdot \frac{1}{3}$ ), you couldn't use exponents. Exponent notation only works with identical factors.

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