Precalculus Exponents and Scientific Notation Study Guide

Q1. Evaluate the expression: $-9^{-2}$

Background

Topic: Negative Exponents

This question tests your understanding of how to evaluate expressions with negative exponents, especially when the base is negative.

Key Terms and Formulas:

• Negative exponent: $a^{-n} = \frac{1}{a^n}$

• Be careful with parentheses and negative signs.

Step-by-Step Guidance

1. Recognize that $-9^{-2}$ means the negative of $9^{-2}$, not $(-9)^{-2}$.

2. Recall the rule for negative exponents: $a^{-n} = \frac{1}{a^n}$.

3. Apply this rule to $9^{-2}$ to rewrite it as a fraction.

4. Now, apply the negative sign in front of the fraction.

Try solving on your own before revealing the answer!

Final Answer: $-\frac{1}{81}$

We have $9^{-2} = \frac{1}{9^2} = \frac{1}{81}$, so $-9^{-2} = -\frac{1}{81}$.

Q2. Evaluate the expression: $\frac{2^9 \cdot 5^{-8}}{2^8 \cdot 5^{-9}}$

Background

Topic: Properties of Exponents

This question tests your ability to simplify expressions using the laws of exponents, including multiplying and dividing powers with the same base and handling negative exponents.

Key Terms and Formulas:

• Product of powers: $a^m \cdot a^n = a^{m+n}$

• Quotient of powers: $\frac{a^m}{a^n} = a^{m-n}$

• Negative exponent: $a^{-n} = \frac{1}{a^n}$

Step-by-Step Guidance

1. Separate the expression into two parts: one with base 2 and one with base 5.

2. For the base 2 part: $\frac{2^9}{2^8} = 2^{9-8}$.

3. For the base 5 part: $\frac{5^{-8}}{5^{-9}} = 5^{-8-(-9)}$.

4. Simplify the exponents for both bases.

5. Combine the results to get a single expression.

Try solving on your own before revealing the answer!

Final Answer: $10$

$2^{9-8} = 2^1 = 2$ and $5^{-8-(-9)} = 5^{1}$, so $2 \times 5 = 10$.

Q3. Evaluate the expression: $\left(\frac{11}{2}\right)^{-2}$

Background

Topic: Negative Exponents with Fractions

This question tests your understanding of how to apply negative exponents to fractions.

Key Terms and Formulas:

• Negative exponent: $a^{-n} = \frac{1}{a^n}$

• For a fraction: $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$

Step-by-Step Guidance

1. Apply the negative exponent rule to the fraction: invert the fraction and make the exponent positive.

2. Rewrite $\left(\frac{11}{2}\right)^{-2}$ as $\left(\frac{2}{11}\right)^2$.

3. Square both the numerator and the denominator.

4. Write the result as a single fraction.

Try solving on your own before revealing the answer!

Final Answer: $\frac{4}{121}$

Inverting and squaring gives $\left(\frac{2}{11}\right)^2 = \frac{4}{121}$.

Q4. Simplify the expression: $\left(\frac{2x^3y}{x^4}\right)^3$

Background

Topic: Exponent Rules and Simplification

This question tests your ability to apply exponent rules to simplify algebraic expressions, including distributing exponents and combining like terms.

Key Terms and Formulas:

• Power of a quotient: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

• Power of a product: $(ab)^n = a^n b^n$

• Exponent rules: $x^m \cdot x^n = x^{m+n}$, $\frac{x^m}{x^n} = x^{m-n}$

Step-by-Step Guidance

1. Apply the exponent of 3 to both the numerator and denominator.

2. Expand the numerator: $2^3 (x^3)^3 y^3$.

3. Expand the denominator: $(x^4)^3$.

4. Simplify the exponents in both numerator and denominator.

5. Write the final expression using positive exponents only.

Try solving on your own before revealing the answer!

Final Answer: $\frac{8y^3}{x^3}$

After simplifying, you get $2^3 = 8$, $x^{3 \times 3} = x^9$, $y^3$, and $x^{4 \times 3} = x^{12}$. So $\frac{8x^9y^3}{x^{12}} = \frac{8y^3}{x^3}$.

Q5. Simplify the expression: $\left(-3x^4\right)^{-2}$

Background

Topic: Negative Exponents and Powers

This question tests your ability to handle negative exponents applied to products and powers.

Key Terms and Formulas:

• Negative exponent: $a^{-n} = \frac{1}{a^n}$

• Power of a product: $(ab)^n = a^n b^n$

• Power of a power: $(x^m)^n = x^{mn}$

Step-by-Step Guidance

1. Apply the negative exponent by taking the reciprocal and making the exponent positive.

2. Rewrite $\left(-3x^4\right)^{-2}$ as $\frac{1}{(-3x^4)^2}$.

3. Expand the denominator: $(-3)^2 (x^4)^2$.

4. Simplify the exponents and coefficients.

5. Write the result as a single fraction with positive exponents.

Try solving on your own before revealing the answer!

Final Answer: $\frac{1}{9x^8}$

$(-3)^2 = 9$ and $(x^4)^2 = x^8$, so the answer is $\frac{1}{9x^8}$.

Q6. Simplify the exponential expression: $\frac{x^3 y^{-3}}{x^{-1} y}$

Background

Topic: Exponent Rules with Variables

This question tests your ability to simplify expressions with both positive and negative exponents in the numerator and denominator.

Key Terms and Formulas:

• Quotient of powers: $\frac{a^m}{a^n} = a^{m-n}$

• Negative exponent: $a^{-n} = \frac{1}{a^n}$

Step-by-Step Guidance

1. For the $x$ terms: $\frac{x^3}{x^{-1}} = x^{3-(-1)}$.

2. For the $y$ terms: $\frac{y^{-3}}{y^1} = y^{-3-1}$.

3. Simplify the exponents for both $x$ and $y$.

4. Write the final answer using only positive exponents.

Try solving on your own before revealing the answer!

Final Answer: $\frac{x^4}{y^4}$

$x^{3-(-1)} = x^4$ and $y^{-3-1} = y^{-4}$, so $\frac{x^4}{y^4}$.

Q7. Simplify the expression: $\frac{3x^5y^4z^2}{x^5y^4z^{-7}}$

Background

Topic: Exponent Rules with Multiple Variables

This question tests your ability to apply exponent rules to expressions with several variables and both positive and negative exponents.

Key Terms and Formulas:

• Quotient of powers: $\frac{a^m}{a^n} = a^{m-n}$

• Negative exponent: $a^{-n} = \frac{1}{a^n}$

Step-by-Step Guidance

1. For each variable, subtract the exponent in the denominator from the exponent in the numerator.

2. For $x$: $x^{5-5}$; for $y$: $y^{4-4}$; for $z$: $z^{2-(-7)}$.

3. Simplify each exponent.

4. Write the result as a product of powers with positive exponents.

Try solving on your own before revealing the answer!

Final Answer: $3z^9$

$x^{0} = 1$, $y^{0} = 1$, $z^{2-(-7)} = z^9$, so the answer is $3z^9$.

Q8. Simplify the expression: $\left(\frac{xy^{-4}z^{-2}}{x^2y^{-5}z^3}\right)^{-2}$

Background

Topic: Negative Exponents and Complex Fractions

This question tests your ability to simplify complex expressions with negative exponents and powers of quotients.

Key Terms and Formulas:

• Power of a quotient: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

• Negative exponent: $a^{-n} = \frac{1}{a^n}$

• Exponent rules: $x^m \cdot x^n = x^{m+n}$, $\frac{x^m}{x^n} = x^{m-n}$

Step-by-Step Guidance

1. First, simplify the inside of the fraction by subtracting exponents for each variable.

2. Apply the outer exponent of $-2$ to each term in the simplified fraction.

3. Remember that raising a power to a negative exponent inverts the base and changes the sign of the exponent.

4. Write the final answer using only positive exponents.

Try solving on your own before revealing the answer!

Final Answer: $\frac{z^{10}x^2}{y^2}$

After simplifying, you get $z^{10}x^2/y^2$.

Q9. Express the number $6,600,000$ in scientific notation.

Background

Topic: Scientific Notation

This question tests your ability to convert a standard number into scientific notation.

Key Terms and Formulas:

• Scientific notation: $a \times 10^n$, where $1 \leq a < 10$ and $n$ is an integer.

Step-by-Step Guidance

1. Identify the first nonzero digit and place the decimal after it to get a number between 1 and 10.

2. Count how many places you move the decimal point to reach this form.

3. Write the number as $a \times 10^n$, where $n$ is the number of places moved.

Try solving on your own before revealing the answer!

Final Answer: $6.6 \times 10^6$

The decimal moves 6 places to the left, so $6,600,000 = 6.6 \times 10^6$.

Q10. Convert the number $0.0000811$ to scientific notation.

Background

Topic: Scientific Notation for Small Numbers

This question tests your ability to express small numbers in scientific notation.

Key Terms and Formulas:

• Scientific notation: $a \times 10^n$, where $1 \leq a < 10$ and $n$ is an integer.

Step-by-Step Guidance

1. Move the decimal point to the right until you have a number between 1 and 10.

2. Count the number of places you moved the decimal; this will be a negative exponent.

3. Write the number as $a \times 10^{-n}$.

Try solving on your own before revealing the answer!

Final Answer: $8.11 \times 10^{-5}$

The decimal moves 5 places to the right, so $0.0000811 = 8.11 \times 10^{-5}$.

Q11. Complete the following table by expressing the equatorial diameters in scientific notation:

• Planet-1: $12,900$ km

• Planet-2: $3,490$ km

• Planet-3: $1,370,000$ km

• Planet-4: $133,000$ km

• Planet-5: $4,600$ km

Background

Topic: Scientific Notation for Large Numbers

This question tests your ability to convert various large numbers into scientific notation.

Key Terms and Formulas:

• Scientific notation: $a \times 10^n$, where $1 \leq a < 10$ and $n$ is an integer.

Step-by-Step Guidance

1. For each number, move the decimal point to create a number between 1 and 10.

2. Count the number of places the decimal moves for each value.

3. Write each as $a \times 10^n$.

Try solving on your own before revealing the answer!

Final Answers:

• Planet-1: $1.29 \times 10^4$ km

• Planet-2: $3.49 \times 10^3$ km

• Planet-3: $1.37 \times 10^6$ km

• Planet-4: $1.33 \times 10^5$ km

• Planet-5: $4.60 \times 10^3$ km

Each number is written in the form $a \times 10^n$.

Q12. Write $1,540,000,000,000$ in scientific notation.

Background

Topic: Scientific Notation for Large Numbers

This question tests your ability to convert a very large number into scientific notation.

Key Terms and Formulas:

• Scientific notation: $a \times 10^n$, where $1 \leq a < 10$ and $n$ is an integer.

Step-by-Step Guidance

1. Move the decimal point to create a number between 1 and 10.

2. Count the number of places the decimal moves.

3. Write the number as $a \times 10^n$.

Try solving on your own before revealing the answer!

Final Answer: $1.54 \times 10^{12}$

The decimal moves 12 places to the left, so $1,540,000,000,000 = 1.54 \times 10^{12}$.

Q13. Convert $0.00000000047$ to scientific notation.

Background

Topic: Scientific Notation for Small Numbers

This question tests your ability to express very small numbers in scientific notation.

Key Terms and Formulas:

• Scientific notation: $a \times 10^n$, where $1 \leq a < 10$ and $n$ is an integer.

Step-by-Step Guidance

1. Move the decimal point to the right until you have a number between 1 and 10.

2. Count the number of places you moved the decimal; this will be a negative exponent.

3. Write the number as $a \times 10^{-n}$.

Try solving on your own before revealing the answer!

Final Answer: $4.7 \times 10^{-10}$

The decimal moves 10 places to the right, so $0.00000000047 = 4.7 \times 10^{-10}$.

Q14. Write $6,150,000,000,000,000,000,000,000$ kilograms in scientific notation.

Background

Topic: Scientific Notation for Very Large Numbers

This question tests your ability to convert extremely large numbers into scientific notation.

Key Terms and Formulas:

• Scientific notation: $a \times 10^n$, where $1 \leq a < 10$ and $n$ is an integer.

Step-by-Step Guidance

1. Move the decimal point to create a number between 1 and 10.

2. Count the number of places the decimal moves.

3. Write the number as $a \times 10^n$.

Try solving on your own before revealing the answer!

Final Answer: $6.15 \times 10^{24}$ kg

The decimal moves 24 places to the left, so $6,150,000,000,000,000,000,000,000 = 6.15 \times 10^{24}$ kg.