Question

In Exercises, evaluate each expression. (a) \(5\left(5^{3}\right)\) (b) \(27^{2 / 3}\) (c) \(64^{3 / 4}\) (d) \(81^{1 / 2}\) (e) \(25^{3 / 2}\) (f) \(32^{2 / 5}\)

Short answer

(a) 625, (b) 9, (c) 16, (d) 9, (e) 125, (f) 4

Step-by-step solution

Evaluate the expression (a)

For \(5\left(5^{3}\right)\), use the order of operations, or BIDMAS / PEMDAS to firstly evaluate the exponent before the multiplication, \(5^3 = 125\). Then multiply this result by 5 to get 625.

Evaluate the expression (b)

For \(27^{2 / 3}\), the fractional exponent signifies both an exponent and a root. Here, the numerator 2 is the power and the denominator 3 is the root. So it is equivalent to \((27^2)^{1/3} = 729^{1/3}\), which equals 9 since the cube root of 729 is 9.

Evaluate the expression (c)

For \(64^{3 / 4}\), again, we have both an exponent and a root due to the fractional exponent. So it is equivalent to \((64^3)^{1/4} = 262144^{1/4}\). The fourth root of 262144 is 16.

Evaluate the expression (d)

For \(81^{1 / 2}\), the one-half exponent signifies a square root. The square root of 81 is 9.

Evaluate the expression (e)

For \(25^{3 / 2}\), we have a cube (from the numerator 3) and a square root (from the denominator 2). So it is equivalent to \((25^3)^{1/2} = 15625^{1/2}\). The square root of 15625 is 125.

Evaluate the expression (f)

For \(32^{2 / 5}\), we have a square (from the numerator 2) and a fifth root (from the denominator 5). So it is equivalent to \((32^2)^{1/5} = 1024^{1/5}\). The fifth root of 1024 is 4.

Key concepts

Exponentiation

Exponentiation is a powerful mathematical operation involving numbers raised to a certain power. It is denoted by a base number, followed by a superscript exponent. For example, in the expression \(5^3\), 5 is the base, and 3 is the exponent. This means you multiply the base, 5, by itself, 3 times:

• 5 × 5 × 5 = 125

In general, the exponent tells you how many times to use the base in a multiplication. Exponentiation is essential in simplifying expressions and solving equations. Understanding exponentiation also lays the groundwork for mastering advanced topics like roots and logarithms.
The main point to remember is that the larger the exponent, the more times you'll repeat the multiplication process. This repeated multiplication can often result in very large numbers.

Roots

Roots are the inverse operation of exponentiation. When you see a fractional exponent, it usually involves finding a root. For example, the expression \(27^{2/3}\) combines both raising to a power and finding a root.

• The denominator (3 in this example) indicates the root that needs to be taken, so you're looking for a cube root.
• The numerator (2 in this example) shows that the number also needs to be squared.

So, 27 raised to the power of 2/3 is equivalent to first squaring 27 (to get 729) and then taking the cube root of 729, which equals 9.
Roots play a key role in a variety of mathematical operations and are particularly useful when working with fractional exponents. This method helps in transforming complex irrational numbers into more manageable forms.

Order of Operations

The order of operations is vital for correctly evaluating mathematical expressions. It's often remembered using the acronym PEMDAS or BIDMAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This order helps you decide which part of a complex expression to solve first.
For instance, in the expression \(5(5^3)\), you first calculate \(5^3 = 125\), then multiply the result by 5 to get 625.

• Always solve exponents before multiplication or division.
• Perform operations inside parentheses before anything else.

By keeping the order of operations in mind, you ensure that you are simplifying or evaluating expressions correctly without ambiguity or error.

Simplifying Expressions

Simplifying expressions means rewriting them in a simpler or more easily understandable form. This involves reducing expressions to their simplest terms. When dealing with exponents and roots, simplifying might involve calculating powers first and then simplifying the resulting number using roots, or vice versa, depending on the expression.
Take, for example, the expression \(64^{3/4}\), which can be simplified by either first reviewing it as \((64^3)^{1/4}\). We square 64 and then take the fourth root of 262144 to simplify it to 16.

• Break down expressions by handling one operation at a time.
• Use properties of exponents and roots to make calculations easier.

Simplifying complex expressions not only makes calculations easier but also aids in understanding the fundamental operations and in solving more complicated algebraic equations.