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 number being multiplied is called the base, and the small number written above and to the right of the base is the exponent or power, which 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\(, which is read as "eight to the fourth power." More generally, if a number \)b\( is multiplied by itself \)n\( times, it can be expressed as \)b^n\(, or "b to the nth power."

Exponent notation not only simplifies writing but also helps in understanding and calculating values. To find the value of an exponential expression, you can expand it back into repeated multiplication. For instance, \)7^2\( (seven squared) means \)7 \times 7\(, which equals 49. The exponent 2 is often called "squared," and similarly, the exponent 3 is called "cubed."

Another example is \)10^3\( (ten cubed), which expands to \)10 \times 10 \times 10\(. Calculating step-by-step, \)10 \times 10 = 100\(, and then \)100 \times 10 = 1,000\(. Thus, \)10^3 = 1,000\(.

For higher exponents, such as \)2^5\(, the process is the same: multiply the base by itself five times: \)2 \times 2 \times 2 \times 2 \times 2\(. Grouping the multiplication can make calculations easier: \)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\(. Therefore, \)2^5 = 32\(.

It is also important to note that if a number has no visible exponent, it is understood to have an exponent of 1. For example, the number 4 can be written as \)4^1$, and both represent the same value.

Understanding exponent notation is fundamental for working efficiently with repeated multiplication and lays the groundwork for more advanced mathematical concepts involving powers and exponents.

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 1/3 by itself four times: 1/3 × 1/3 × 1/3 × 1/3. When multiplying fractions, multiply the numerators together and the denominators together. Starting with the first two fractions, 1/3 × 1/3 equals 1/9 because 1 × 1 = 1 and 3 × 3 = 9. Next, multiply 1/9 × 1/3 to get 1/27, and finally multiply 1/27 × 1/3 to arrive at 1/81. Thus, (\frac{1}{3})^4 = \frac{1}{81}.

Recognizing patterns can simplify calculations. For instance, since (\frac{1}{3})^2 = \frac{1}{9}, you can rewrite (\frac{1}{3})^4 as (\frac{1}{3})^2 \times (\frac{1}{3})^2 = \frac{1}{9} \times \frac{1}{9} = \frac{1}{81}, reducing the number of multiplication steps. This approach highlights the importance of understanding exponent rules and fraction multiplication to efficiently solve problems involving fractional exponents.

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.

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 more efficient and easier to understand.

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 a product and calculating the result.

What special terms are used for exponents 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 are commonly used in mathematics and science, making communication clearer and more concise. Recognizing these terms helps students quickly understand and work with exponential expressions without always needing to say "to the power of 2" or "to the power of 3."

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

To evaluate $2^5$, you multiply 2 by itself five times: 2 × 2 × 2 × 2 × 2. You can group the multiplication to make it easier: 2 × 2 = 4, another 2 × 2 = 4, so now multiply 4 × 4 = 16, and finally 16 × 2 = 32. Therefore, 2 to the fifth power equals 32. This method of expanding the exponent into repeated multiplication helps you calculate the value step-by-step.

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

When a number has no exponent written, it implies 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. Understanding this concept is important because it helps when working with algebraic expressions and polynomials, where the exponent of 1 is often not written explicitly but is always assumed.