Solve for x in (3/10)^x: Complete the Exponential Expression

Q: Why can't I just leave it as (2×5)^x in the denominator?

While $ (2 \times 5)^x $ is mathematically correct, the question asks for the complete expression. You need to apply the product power rule: $ (ab)^n = a^n \times b^n $ to get $ 2^x \times 5^x $.

Q: What's the difference between (2×5)^x and 2^x×5^x?

They're mathematically equal but represent different stages of simplification! $ (2 \times 5)^x = 2^x \times 5^x $ because of the product power rule. The expanded form shows each base raised to the power separately.

Q: How do I remember the fraction exponent rule?

Think of it as "exponent goes to both floors" - the numerator and denominator each get the exponent! So $ \left(\frac{a}{b}\right)^n $ becomes $ \frac{a^n}{b^n} $.

Q: Can I simplify 2×5 to 10 first?

You could, but that would give you $ \frac{3^x}{10^x} $, which is different from the required answer $ \frac{3^x}{2^x \times 5^x} $. The question wants you to keep the factors separate.

Q: What if I get confused about which rule to apply when?

Start with the quotient power rule first: $ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $. Then, if the denominator has multiplication, apply the product power rule: $ (ab)^n = a^n b^n $.

Exponent Distribution with Fractional Bases

Insert the corresponding expression:

$\left(\frac{3}{2\times5}\right)^x=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:10 Let's simplify this problem together.

00:14 Remember, if you have a fraction with a power N, both the top and bottom get the same power N.

00:20 So each part of the fraction will be raised to the power of N.

00:25 Now let's try this out with our example.

00:28 When you have a product with a power N, it means each factor gets that power.

00:34 Let’s apply this law to see what happens.

00:38 We'll use this rule in our calculation now.

00:43 And that gives us our solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(\frac{3}{2\times5}\right)^x=$

Step-by-step solution

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

Step 1: Identify the given expression.
Step 2: Apply the exponent rule $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ to distribute $x$ to numerator and denominator.
Step 3: Simplify the denominator expression by distributing exponent $x$ to each factor.

Now, let's work through each step:
Step 1: We have the original expression $\left(\frac{3}{2 \times 5}\right)^x$ .
Step 2: Apply the rule to get $\frac{3^x}{(2 \times 5)^x}$ .
Step 3: Expand the denominator: $(2 \times 5)^x = 2^x \times 5^x$ . This leads us to $\frac{3^x}{2^x \times 5^x}$ .

Therefore, the solution to the problem is $\frac{3^x}{2^x \times 5^x}$ .

Final Answer

$\frac{3^x}{2^x\times5^x}$

Key Points to Remember

Essential concepts to master this topic

Rule: Apply $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ to distribute exponents
Technique: Expand $(2 \times 5)^x$ to $2^x \times 5^x$
Check: Verify by substituting a test value like x=2 into both forms ✓

Common Mistakes

Avoid these frequent errors

Leaving the denominator as (2×5)^x without expanding
Don't write $\frac{3^x}{(2 \times 5)^x}$ as your final answer! This doesn't fully apply the exponent distribution rule and misses a crucial simplification step. Always expand $(2 \times 5)^x$ to $2^x \times 5^x$ using the product power rule.

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 leave it as (2×5)^x in the denominator?

While $(2 \times 5)^x$ is mathematically correct, the question asks for the complete expression. You need to apply the product power rule: $(ab)^n = a^n \times b^n$ to get $2^x \times 5^x$ .

What's the difference between (2×5)^x and 2^x×5^x?

They're mathematically equal but represent different stages of simplification! $(2 \times 5)^x = 2^x \times 5^x$ because of the product power rule. The expanded form shows each base raised to the power separately.

How do I remember the fraction exponent rule?

Think of it as "exponent goes to both floors" - the numerator and denominator each get the exponent! So $\left(\frac{a}{b}\right)^n$ becomes $\frac{a^n}{b^n}$ .

Can I simplify 2×5 to 10 first?

You could, but that would give you $\frac{3^x}{10^x}$ , which is different from the required answer $\frac{3^x}{2^x \times 5^x}$ . The question wants you to keep the factors separate.

What if I get confused about which rule to apply when?

Start with the quotient power rule first: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ . Then, if the denominator has multiplication, apply the product power rule: $(ab)^n = a^n b^n$ .

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