Solve: (17^-3 × 17^3x)/17 - 17x Using Exponent Rules

Q: Why can't I combine the two terms in the final answer?

The terms $ 17^{3x-4} $ and $ -17x $ have different bases! One has base 17 with an exponent, the other is just 17 times x. You can only combine terms with identical bases and exponents.

Q: Do I apply exponent rules to both terms in the expression?

No! Only apply exponent rules to the fraction part: $ \frac{17^{-3} \cdot 17^{3x}}{17} $. The term $ -17x $ stays as is because it's not part of the fraction.

Q: How do I know when to use the product rule vs quotient rule?

Use the product rule when multiplying (like $ 17^{-3} \cdot 17^{3x} $). Use the quotient rule when dividing (like $ \frac{17^{something}}{17} $). Look for multiplication or division symbols!

Q: Why is 17 in the denominator written as 17¹?

Any number without an exponent has an implied exponent of 1. So $ 17 = 17^1 $. This lets us use the quotient rule: $ \frac{17^{-3+3x}}{17^1} = 17^{(-3+3x)-1} $.

Q: Can I simplify 3x - 4 in the exponent?

No! The expression $ 3x - 4 $ in the exponent is already simplified. You can only combine like terms, and 3x and -4 are not like terms.

Exponent Rules with Fraction Simplification

$\frac{17^{-3}\cdot17^{3x}}{17}-17x=\text{?}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:09 In order to eliminate a negative exponent

00:12 We'll invert the numerator and the denominator in order that the exponent will become positive

00:15 We'll apply this formula to our exercise, and then convert from a fraction to a exponent

00:20 When multiplying powers with equal bases

00:23 The exponent of the result equals the sum of the exponents

00:28 We'll apply this formula to our exercise, we'll then add up the exponents

00:41 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$\frac{17^{-3}\cdot17^{3x}}{17}-17x=\text{?}$

Step-by-step solution

Let's deal with the first term in the problem, which is the fraction,

For this, we'll recall two laws of exponents:

a. The law of exponents for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$ b. The law of exponents for division between terms with identical bases:

$\frac{a^m}{a^n}=a^{m-n}$ Let's apply these laws of exponents to the problem:

$\frac{17^{-3}\cdot17^{3x}}{17}-17x=\frac{17^{-3+3x}}{17}-17x=17^{-3+3x-1}-17x=17^{3x-4}-17x$ where in the first stage we'll apply the law of exponents mentioned in 'a' above to the fraction's numerator, and in the next stage we'll apply the law of exponents mentioned in 'b' to the resulting expression, then we'll simplify the expression.

Therefore, the correct answer is answer a.

Final Answer

$17^{3x-4}-17x$

Key Points to Remember

Essential concepts to master this topic

Product Rule: When multiplying same bases, add exponents: $a^m \cdot a^n = a^{m+n}$
Quotient Rule: When dividing same bases, subtract exponents: $\frac{a^m}{a^n} = a^{m-n}$
Verification: Check that $17^{3x-4} - 17x$ cannot be simplified further ✓

Common Mistakes

Avoid these frequent errors

Adding exponents when dividing instead of subtracting
Don't add exponents for $\frac{17^{-3+3x}}{17}$ = $17^{-3+3x+1}$ ! Division means subtraction of exponents, not addition. Always subtract the denominator's exponent: $17^{-3+3x-1} = 17^{3x-4}$ .

Practice Quiz

Test your knowledge with interactive questions

Simplify the following equation:

$$ $4^5\times4^5=$

FAQ

Everything you need to know about this question

Why can't I combine the two terms in the final answer?

The terms $17^{3x-4}$ and $-17x$ have different bases! One has base 17 with an exponent, the other is just 17 times x. You can only combine terms with identical bases and exponents.

Do I apply exponent rules to both terms in the expression?

No! Only apply exponent rules to the fraction part: $\frac{17^{-3} \cdot 17^{3x}}{17}$ . The term $-17x$ stays as is because it's not part of the fraction.

How do I know when to use the product rule vs quotient rule?

Use the product rule when multiplying (like $17^{-3} \cdot 17^{3x}$ ). Use the quotient rule when dividing (like $\frac{17^{something}}{17}$ ). Look for multiplication or division symbols!

Why is 17 in the denominator written as 17¹?

Any number without an exponent has an implied exponent of 1. So $17 = 17^1$ . This lets us use the quotient rule: $\frac{17^{-3+3x}}{17^1} = 17^{(-3+3x)-1}$ .

Can I simplify 3x - 4 in the exponent?

No! The expression $3x - 4$ in the exponent is already simplified. You can only combine like terms, and 3x and -4 are not like terms.

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Solve: (17^-3 × 17^3x)/17 - 17x Using Exponent Rules

Exponent Rules with Fraction Simplification

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why can't I combine the two terms in the final answer?

Do I apply exponent rules to both terms in the expression?

How do I know when to use the product rule vs quotient rule?

Why is 17 in the denominator written as 17¹?

Can I simplify 3x - 4 in the exponent?

🌟 Unlock Your Math Potential

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