Solving expressions with exponents involves understanding and applying rules such as the product of powers, quotient of powers, and power of a power. For instance, a^m × aⁿ = a^m+n and (a^m)ⁿ = a^mn. These properties simplify complex calculations. For example, (2³)² × 2⁻⁴ simplifies to 2⁶⁻⁴ = 2² = 4. Understanding these rules helps in efficiently handling exponential expressions and equations.

In this article, we will discuss various different solved and unsolved examples solving Expressions with Exponents.

What are Expressions?

Expressions in mathematics are combinations of numbers, variables, and operations (such as addition, subtraction, multiplication, and division) grouped together to represent a value or relationship.

An expression can be as simple as a single number (e.g., 5) or a variable (e.g., x), or it can be more complex, involving multiple terms and operations (e.g., 3x + 2 or (a+b)/c.

Note: Expressions do not include equality signs, distinguishing them from equations.

Expressions with Exponents

Expressions with exponents are mathematical expressions that include numbers or variables raised to a power, known as an exponent. For example, in 2³, 2 is the base, and 3 is the exponent, meaning 2 × 2 × 2 = 8.

How to Solve Expressions with Exponents?

To simplify or solve expressions with exponents we can use the following steps:

Step 1: Identify the Base and Exponent.

Step 2: Apply the Rules of Exponent such as:

• Product of Powers: a^m × aⁿ = a^m+n
• Quotient of Powers: a^m/aⁿ = a^m-n
• Power of a Power: (a^m)ⁿ = a^mn
• Zero Exponent Rule: a⁰ = 1 (for any non-zero a)
• Negative Exponents: a^-m = 1/a^m
• Power of a Product: (ab)^m = a^m × b^m
• Power of a Quotient: (a/b)^m = a^m/b^m

Step 3: Simplify the complex expression.

Solved Examples for Expressions with Exponents

Example 1: Simplify the expression 3⁴ × 3².

Solution:

Using the product of powers rule:

a^m × aⁿ = a^{m + n}

3⁴ × 3² = 3⁴⁺² = 36

Calculating 36:

3⁶ = 729

So, 3⁴ × 3² = 729.

Example 2: Simplify the expression 5⁷/5³.

Solution:

Using the quotient of powers rule:

a^m/aⁿ = a^m-n

5⁷/5³ = 5^7-3 = 5⁴

Calculating 5⁴:

5⁴ = 625

So, 5⁷/5³ = 625.

Example 3: Simplify the expression (2³)⁴.

Solution:

Using the power of a power rule:

(a^m)ⁿ = a^mn

(2³)⁴ = 2^{3 × 4} = 2¹²

Calculating 2¹²:

2¹² = 4096

So, (2³)⁴ = 4096.

Example 4: Simplify the expression 4^-2.

Solution:

Using the negative exponent rule:

a^-m = 1/a^m

4^-2 = 1/4²

Calculating 4²:

4² = 16

So, 4^-2 = 1/16

Example 5: Simplify the expression (2⁵ × 2^-3)/2².

Solution:

First, apply the product of powers rule:

(2⁵ × 2^-3) = 2^{5+ (-3)} = 2^5-3 = 2²

Then, apply the quotient of powers rule:

2²/ 2² = 2^2-2 = 2⁰

Using the zero exponent rule:

2⁰ = 1

So, (2⁵ × 2^-3)/2² = 1.

Example 6: Simplify the expression (3 × 4)².

Solution:

Using the power of a product rule:

(ab)^m = a^m × b^m

(3 × 4)² = 3² × 4²

Calculating each term:

3² = 9

4² = 16

So,

9 × 16 = 144

Therefore, (3 × 4)² = 144.

Example 7: Simplify the expression 2³/4.

Solution:

First, express 4 as 2²:

4 = 2²

Then rewrite the expression:

2³/4 = 2³/2²

Using the quotient of powers rule:

a^m/aⁿ = a^m-n

2³/2² = 2^3-2 = 2¹ = 2

So, 2³/4 = 2.

Example 8: Simplify the expression 16^3/4.

Solution:

Using the fractional exponent rule:

a^m/n = [Tex]\sqrt[n]{a^m}[/Tex]

[Tex]16^{3/4} = \sqrt[4]{16^3}[/Tex]

Calculating 16³:

16³ = 16 × 16 × 16 = 4096

Then take the fourth root of 4096:

[Tex]\sqrt[4]{4096} [/Tex]= 8

So, 16^3/4 = 8.

Example 9: Simplify the expression 27^-2/3.

Solution:

Using the negative exponent and fractional exponent rules:

[Tex]a{-m/n} = \frac{1}{a{m/n}}[/Tex], and

a^m/n = [Tex]\sqrt[n]{a^m}[/Tex]

[Tex]27^{-2/3} = \frac{1}{27^{2/3}} = \frac{1}{\sqrt[3]{27^2}}[/Tex]

Calculating 27²:

27² = 729

Then take the cube root of 729:

[Tex]\sqrt[3]{729} = 9[/Tex]

So, [Tex]\frac{1}{27^{2/3}} = \frac{1}{9}[/Tex]

Therefore, 27^-2/3 = 1/9

Example 10: Simplify the expression (2/3)⁴.

Solution:

Using the power of a quotient rule:

[Tex]\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}[/Tex]

[Tex]\left(\frac{2}{3}\right)^4 = \frac{2^4}{3^4}[/Tex]

Calculating each term:

2⁴ = 16

3⁴ = 81

So, [Tex]\frac{2^4}{3^4} = \frac{16}{81}[/Tex]

Practice Problems for Expressions with Exponents

Problem 1: Simplify the expression 4³ × 4⁵.

Problem 2: Simplify the expression 6⁸/6³.

Problem 3: Simplify the expression (5²)³.

Problem 4: Simplify the expression 7^-2.

Problem 5: Simplify the expression (2 × 5)³.

Problem 6: Simplify the expression 81^3/4.

Problem 7: Simplify the expression 32^-3/5.

Problem 8: Simplify the expression (10⁴ × 10^-4)⁰.

Problem 9: Simplify the expression [Tex] \left(\frac{3}{4}\right)^2[/Tex].

Problem 10: Simplify the expression [Tex]\frac{2^5 × 2^{-3}}{2^2}[/Tex].

FAQs: Expressions with Exponents

Define exponent.

An exponent refers to the number that indicates how many times the base number is multiplied by itself.

What are the basic rules of exponents?

The basic rules of exponents include:

• Product of Powers: a^m × aⁿ = a^m+n
• Quotient of Powers: a^m/aⁿ = a^m-n
• Power of a Power: (a^m)ⁿ = a^mn
• Zero Exponent Rule: a⁰ = 1 (for any non-zero a)
• Negative Exponents: a^-m = 1/a^m

How do you simplify expressions with exponents?

Simplifying expressions with exponents involves applying the relevant rules of exponents.

What does a negative exponent signify?

A negative exponent signifies the reciprocal of the base raised to the positive exponent.