Rewrite the product as an exponential expression.(29)×(29)×(29)×(29)×(29)\left(\frac29\right)\times\left(\frac29\right)\times\left(\frac29\right)\times\left(\frac29\right)\times\left(\frac29\right)

(29)5\left(\frac29\$$right)^5$$

$$5(29)5^{\left(\frac29\right)}$$

(29)29\left(\frac29\$$right)^{\frac29}$$

4. The Real Number System

Integers & the Order of Operations

4. The Real Number System

Integers & the Order of Operations: Videos & Practice Problems

Video Lessons

Concept

Intro to Exponents

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Intro to Exponents Video Summary

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."

This notation can be generalized for any base \)b$ and exponent $n$, written as $b^n$, meaning $b$ multiplied by itself $n\( times. Understanding how to convert between repeated multiplication and exponent notation is essential for simplifying expressions and performing calculations efficiently.

For instance, \)7^2$ (seven squared) means \(7 \times 7$, which equals 49. The exponent 2 is often called "squared" because it represents the area of a square with side length 7. Similarly, \)10^3$ (ten cubed) means \(10 \times 10 \times 10$, which equals 1,000. The exponent 3 is called "cubed" because it represents the volume of a cube with side length 10.

For exponents beyond 3, such as \)2^5$, the expression means multiplying 2 by itself five times: \(2 \times 2 \times 2 \times 2 \times 2$. Calculating this step-by-step, $2 \times 2 = 4$, then $4 \times 2 = 8$, $8 \times 2 = 16$, and finally $16 \times 2 = 32$. Thus, $2^5 = 32$.

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$, which simply means 4 itself.

Mastering exponent notation not only simplifies the representation of repeated multiplication but also lays the foundation for more advanced mathematical concepts such as powers, roots, and exponential functions.

Problem

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

$7 cubed$

$3 to the seventh power$

$3^3$

$7^7$

Problem

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

$5 raised to the two ninths power$

${(\frac{2}{9})}^{\frac{2}{9}}$

$5 to the fifth power$

${(\frac{2}{9})}^{5}$

Problem

Evaluate the following.
$13 to the first power$

$13$

$169$

$0$

$1$

Problem

Evaluate the following.
$7 cubed$

$21$

$49$

$343$

$140.6$

Problem

Evaluate the following.
$2 to the eighth power$

$16$

$256$

$5962$

$64$

Example

Intro to Exponents Example 1

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Intro to Exponents Example 1 Video Summary

When evaluating an exponential expression with a fractional base, such as one third raised to the fourth power, the process involves multiplying the base by itself as many times as indicated by the exponent. In this case, one third to the fourth power means multiplying one third four times:

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

Multiplying fractions requires multiplying 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, or:

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

Recognizing patterns can simplify calculations. For example, 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:

$\left(a^m\right)^n = a^{m \times n}$

and the multiplication rule for exponents with the same base:

$a^m \times a^n = a^{m+n}$

Understanding how to multiply fractional bases raised to powers is essential in algebra and helps build a strong foundation for working with rational exponents and more complex expressions.

Example

Intro to Exponents Example 2

Video duration:

51s

Intro to Exponents Example 2 Video Summary

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.

Example

Intro to Exponents Example 3

Video duration:

50s

Intro to Exponents Example 3 Video Summary

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 with 1.3 as the base and 3 as the exponent, written as \$1.3^3$. This notation simplifies repeated multiplication and is fundamental in understanding powers and exponents in mathematics.