Evaluating Exponents: Videos & Practice Problems

Exponent notation simplifies repeated multiplication by expressing a base raised to a power, such as $b^n$, where b is the base and n the exponent. Special terms like "squared" for power two and "cubed" for power three help describe exponential expressions. Understanding how to expand and calculate values, like $2^5=\; 32$, is essential. Recognizing that a number without an exponent implies a power of one aids in working with polynomials, monomials, and terms in standard form.

Intro to Exponents

When a number is multiplied by itself multiple times, writing out the entire multiplication can be cumbersome. Exponent notation offers a compact and efficient way to express this repeated multiplication. In exponent notation, the base is the number being multiplied, and the exponent (or power) indicates how many times the base is multiplied by itself. For example, the expression \$8 \times 8 \times 8 \times 8\( can be written as \)8^4\(, which is read as "eight to the fourth power."

This notation applies universally: if a number b is multiplied by itself n times, it is written as \)b^n\(, read as "b to the nth power." Understanding how to convert between repeated multiplication and exponent notation is essential for simplifying expressions and performing calculations efficiently.

To evaluate exponential expressions, you can expand them back into repeated multiplication. For instance, \)7^2\( (seven squared) means \)7 \times 7\(, which equals 49. The exponent 2 is often called "squared," a special term used for this power.

Similarly, \)10^3\( (ten cubed) represents \)10 \times 10 \times 10\(. Calculating this step-by-step, \)10 \times 10 = 100\(, and then \)100 \times 10 = 1,000\(. The exponent 3 is commonly referred to as "cubed."

For higher exponents, such as \)2^5\(, you multiply the base five times: \)2 \times 2 \times 2 \times 2 \times 2\(. Grouping the multiplication can simplify the process: \)2 \times 2 = 4\(, another \)2 \times 2 = 4\(, so the expression becomes \)4 \times 4 \times 2\(. Then, \)4 \times 4 = 16\(, and \)16 \times 2 = 32\(. Thus, \)2^5 = 32\(.

It is important to note that if a number has no visible exponent, it is understood to have an exponent of one. For example, the number 4 is equivalent to \)4^1$. This concept is fundamental when working with exponents in more complex expressions.

Mastering exponent notation not only streamlines mathematical writing but also enhances the ability to work with powers and roots in algebra and beyond.

Rewrite the product as an exponential expression.
$3\times 3\times 3\times 3\times 3\times 3\times 3$

• A. $7^3$
• B. $3^7$
• C. $3^3$
• D. $7^7$

Answer: (B)

Rewrite the product as an exponential expression.
$\left(\frac{2}{9}\right)\times \left(\frac{2}{9}\right)\times \left(\frac{2}{9}\right)\times \left(\frac{2}{9}\right)\times \left(\frac{2}{9}\right)$

• A. $5^{\left(\frac{2}{9}\right)}$
• B. ${\left(\frac{2}{9}\right)}^{\frac{2}{9}}$
• C. $5^5$
• D. ${\left(\frac{2}{9}\right)}^5$

Answer: (D)

Evaluate the following.
${13}^1$

• A. $13$
• B. $169$
• C. $0$
• D. $1$

Answer: (A)

Evaluate the following.
$7^3$

• A. $21$
• B. $49$
• C. $343$
• D. $140.6$

Answer: (C)

Evaluate the following.
$2^8$

• A. $16$
• B. $256$
• C. $5962$
• D. $64$

Answer: (B)

Intro to Exponents Example 1

Evaluating exponential expressions with fractional bases follows the same fundamental principle as with whole numbers: multiply the base by itself as many times as indicated by the exponent. For example, to calculate one third raised to the fourth power, you multiply one third by itself four times, expressed as:

\(\left(\frac{1}{3}\right)^4 = \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3}\)

When multiplying fractions, multiply the numerators together and the denominators together. Starting with the first two fractions:

\(\frac{1}{3} \times \frac{1}{3} = \frac{1 \times 1}{3 \times 3} = \frac{1}{9}\)

Next, multiply this result by the third fraction:

\(\frac{1}{9} \times \frac{1}{3} = \frac{1 \times 1}{9 \times 3} = \frac{1}{27}\)

Finally, multiply by the last fraction:

\(\frac{1}{27} \times \frac{1}{3} = \frac{1 \times 1}{27 \times 3} = \frac{1}{81}\)

Thus, one third to the fourth power equals one over eighty-one:

\(\left(\frac{1}{3}\right)^4 = \frac{1}{81}\)

Recognizing patterns can simplify calculations. For instance, since \(\frac{1}{3} \times \frac{1}{3} = \frac{1}{9}\), you can rewrite the expression as:

\(\left(\frac{1}{3}\right)^4 = \left(\frac{1}{3} \times \frac{1}{3}\right) \times \left(\frac{1}{3} \times \frac{1}{3}\right) = \frac{1}{9} \times \frac{1}{9} = \frac{1}{81}\)

This approach leverages the property of exponents that states \(a^m \times a^n = a^{m+n}\), allowing you to break down complex multiplications into simpler steps. Understanding and applying these patterns enhances efficiency and deepens comprehension of exponential operations involving fractions.

Intro to Exponents Example 2

To evaluate the expression seven fourths squared, written as \(\left(\frac{7}{4}\right)^2\), you multiply the fraction by itself. This means calculating \(\frac{7}{4} \times \frac{7}{4}\). When multiplying fractions, multiply the numerators together and the denominators together. Here, the numerator is \$7 \times 7 = 49\(, and the denominator is \)4 \times 4 = 16\(. Therefore, \)\left(\frac{7}{4}\right)^2 = \frac{49}{16}$. This process demonstrates how to raise a fraction to a power by repeated multiplication, which is a fundamental concept in working with exponents and rational numbers.

Intro to Exponents Example 3

When expressing repeated multiplication of the same number using exponents, the base is the number being multiplied, and the exponent indicates how many times it is multiplied by itself. For example, the expression 1.3 × 1.3 × 1.3 can be rewritten as an exponential expression where 1.3 is the base and 3 is the exponent. This is written as \$1.3^3$, which means 1.3 raised to the power of 3. Using exponents simplifies expressions by condensing repeated multiplication into a more compact form, making calculations and algebraic manipulations more efficient and easier to understand.

Here’s what students ask on this topic:

What is exponent notation and how does it simplify repeated multiplication?

Exponent notation is a compact way to express repeated multiplication of the same number. Instead of writing a number multiplied by itself multiple times, we use a base and an exponent. The base is the number being multiplied, and the exponent tells us how many times to multiply the base by itself. For example, $8^4$ means 8 multiplied by itself 4 times: 8 × 8 × 8 × 8. This notation simplifies writing and working with large repeated multiplications, making calculations and algebraic expressions easier to handle.

How do you evaluate an exponential expression like 7 squared or 10 cubed?

To evaluate an exponential expression, rewrite it as repeated multiplication based on the exponent. For example, 7 squared ($7^2$) means 7 × 7, which equals 49. Similarly, 10 cubed (${10}^3$) means 10 × 10 × 10. Multiplying step-by-step: 10 × 10 = 100, then 100 × 10 = 1,000. So, 10 cubed equals 1,000. This process helps you find the value of any exponential expression by expanding it into multiplication and calculating the result.

What special terms are used for exponents like 2 and 3, and why are they important?

Exponents 2 and 3 have special names: exponent 2 is called "squared" and exponent 3 is called "cubed." For example, $7^2$ is read as "7 squared," and ${10}^3$ is read as "10 cubed." These terms are important because they provide a quick and clear way to describe common powers, especially in geometry and algebra. Squared often relates to area calculations, while cubed relates to volume. Recognizing these terms helps in understanding and communicating mathematical concepts efficiently.

How do you evaluate an expression like 2 to the fifth power?

To evaluate $2^5$, multiply 2 by itself five times: 2 × 2 × 2 × 2 × 2. You can group the multiplication to simplify: 2 × 2 = 4, another 2 × 2 = 4, so now multiply 4 × 4 = 16, then multiply 16 × 2 = 32. Therefore, 2 to the fifth power equals 32. This step-by-step multiplication helps you find the value of any exponent by breaking it down into manageable parts.

What does it mean when a number has no exponent written, like the number 4?

When a number has no exponent written, it is understood to have an exponent of 1. For example, the number 4 is the same as $4^1$. This means 4 is multiplied by itself only once, which is just 4. Recognizing this helps when working with algebraic expressions and polynomials, as it clarifies that every term has an exponent, even if it is not explicitly shown.