Solve (ax)^(-t): Working with Negative Exponents and Multiple Variables

Q: Why doesn't the negative exponent make the answer negative?

Great question! A negative exponent means "reciprocal" or "flip to denominator" - it has nothing to do with making numbers negative. Think of $ 2^{-3} = \frac{1}{2^3} = \frac{1}{8} $, which is positive!

Q: Do I apply the exponent to each variable separately?

Yes! When you have $ (ax)^{-t} $, the exponent distributes to both a and x. So you get $ a^{-t} \times x^{-t} $.

Q: Can I leave my answer with negative exponents?

While mathematically correct, it's usually better to convert to positive exponents using fractions. So $ a^{-t} \times x^{-t} $ becomes $ \frac{1}{a^t \times x^t} $.

Q: What if the exponent was positive instead?

If you had $ (ax)^t $, you'd get $ a^t \times x^t $ - no fractions needed! The negative exponent is what creates the reciprocal or fraction form.

Q: How do I remember the power of a product rule?

Think: "Exponents distribute!" Whatever's outside the parentheses applies to everything inside. Like $ (2 \times 3)^2 = 2^2 \times 3^2 = 4 \times 9 = 36 $.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's simplify this problem together.

00:10 Remember, with exponents, if you see a negative exponent, like A to the power of negative N,

00:16 it means we take the reciprocal. So, it's one over A, raised to the power of N.

00:22 Now, let's use this idea to solve our exercise.

00:26 First, change it to the reciprocal.

00:29 Then, raise it to N, but remember, N is positive this time.

00:34 Also, when a whole product is raised to a power, say N,

00:39 each part of the product gets raised to that power of N.

00:44 We'll use this for our problem too.

00:47 And there you have it, that's how we find the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Solve the following equation :

$\left(a\times x\right)^{-t}=$

2

Step-by-step solution

To solve this problem, we'll employ the following strategy:

Step 1: Apply the power of a product rule: $\left(a \times x\right)^{-t} = a^{-t} \times x^{-t}$ .
Step 2: Apply the negative exponent rule: $a^{-t} = \frac{1}{a^t}$ and $x^{-t} = \frac{1}{x^t}$ .

Here's how we do it step by step:
Step 1: We have $\left(a \times x\right)^{-t}$ . By the power of a product rule, this rewrites as $a^{-t} \times x^{-t}$ .
Step 2: Apply the negative exponent rule to each part:
$a^{-t} = \frac{1}{a^t}$ and $x^{-t} = \frac{1}{x^t}$ .
Therefore, $a^{-t} \times x^{-t} = \frac{1}{a^t} \times \frac{1}{x^t} = \frac{1}{a^t \times x^t}$ .

Thus, the solution to the equation $\left(a \times x\right)^{-t}$ is $\frac{1}{a^t \times x^t}$ .

3

Final Answer

$\frac{1}{a^t\times x^t}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: Apply exponent to each factor in parentheses separately
Technique: $(ax)^{-t} = a^{-t} \times x^{-t} = \frac{1}{a^t} \times \frac{1}{x^t}$
Check: Final answer should have positive exponents in denominator only ✓

Common Mistakes

Avoid these frequent errors

Adding a negative sign to the front of the expression
Don't write $(ax)^{-t} = -a^t \times x^t$ = wrong negative result! The negative exponent means reciprocal, not negative number. Always remember that negative exponents create fractions, never negative signs in front.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why doesn't the negative exponent make the answer negative?

+

Great question! A negative exponent means "reciprocal" or "flip to denominator" - it has nothing to do with making numbers negative. Think of $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$ , which is positive!

Do I apply the exponent to each variable separately?

+

Yes! When you have $(ax)^{-t}$ , the exponent distributes to both a and x. So you get $a^{-t} \times x^{-t}$ .

Can I leave my answer with negative exponents?

+

While mathematically correct, it's usually better to convert to positive exponents using fractions. So $a^{-t} \times x^{-t}$ becomes $\frac{1}{a^t \times x^t}$ .

What if the exponent was positive instead?

+

If you had $(ax)^t$ , you'd get $a^t \times x^t$ - no fractions needed! The negative exponent is what creates the reciprocal or fraction form.

How do I remember the power of a product rule?

+

Think: "Exponents distribute!" Whatever's outside the parentheses applies to everything inside. Like $(2 \times 3)^2 = 2^2 \times 3^2 = 4 \times 9 = 36$ .

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Solve (ax)^(-t): Working with Negative Exponents and Multiple Variables

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