Problem 86
Question
Evaluate each expression without using a calculator. $$ 27^{\frac{1}{3}} $$
Step-by-Step Solution
Verified Answer
The value of \(27^{\frac{1}{3}}\) is 3.
1Step 1: Understanding the Cube Root
The cube root of a number can be found by determining what number cube equals to the original number. In this case, the equation to be solved is \(x^3 = 27\).
2Step 2: Solving for Cube Root
Find a number \(x\) such that when \(x\) is raised to the power of 3, it equals 27. The number that meets this criterion is 3, because \(3^3 = 27\). Therefore, \(x = 3\).
Key Concepts
Exponents and RadicalsSolving Cube RootsAlgebraic Expressions
Exponents and Radicals
Exponents and radicals are fundamental concepts in algebra that relate to how we represent numbers in various forms. An exponent indicates how many times a number, called the base, is multiplied by itself. For instance, in the expression \(4^2\), 4 is the base and 2 is the exponent, meaning \(4^2 = 4 \times 4 = 16\).
Radicals, on the other hand, involve finding roots of numbers. A radical can undo the action of an exponent. The most common radicals are square roots, but cube roots, fourth roots, and so on also exist. The cube root, which is the focus of our problem, extracts the number that, when cubed (raised to the third power), gives the original number. It is represented by the radical symbol followed by a three, as in \(\sqrt[3]{x}\), which means 'the cube root of x'. When you see a fractional exponent such as \(x^{\frac{1}{3}}\), it means the same as taking the cube root of x.
Radicals, on the other hand, involve finding roots of numbers. A radical can undo the action of an exponent. The most common radicals are square roots, but cube roots, fourth roots, and so on also exist. The cube root, which is the focus of our problem, extracts the number that, when cubed (raised to the third power), gives the original number. It is represented by the radical symbol followed by a three, as in \(\sqrt[3]{x}\), which means 'the cube root of x'. When you see a fractional exponent such as \(x^{\frac{1}{3}}\), it means the same as taking the cube root of x.
Solving Cube Roots
Solving cube roots is a process of identifying a number that, when raised to the third power, yields the given number. It requires a good understanding of the multiplication table or, for larger numbers, some trial and error or approximation methods.
For instance, in our example \(27^{\frac{1}{3}}\), we're looking for a number which, when cubed, gives 27. To solve this, you can recall the cube of smaller integers or notice patterns in the multiplication table that will help you find that \(3^3 = 27\). Thus, the cube root of 27 is 3, and this is written as \(\sqrt[3]{27} = 3\). Practicing with various numbers helps build familiarity with common cubes, which makes solving cube roots an easier task.
For instance, in our example \(27^{\frac{1}{3}}\), we're looking for a number which, when cubed, gives 27. To solve this, you can recall the cube of smaller integers or notice patterns in the multiplication table that will help you find that \(3^3 = 27\). Thus, the cube root of 27 is 3, and this is written as \(\sqrt[3]{27} = 3\). Practicing with various numbers helps build familiarity with common cubes, which makes solving cube roots an easier task.
Algebraic Expressions
An algebraic expression is a combination of numbers, variables, and mathematical operators like addition, subtraction, multiplication, and division. Expressions can represent general mathematical relationships and can be simplified or manipulated according to algebraic rules.
In the context of cube roots and radicals, the expression \(27^{\frac{1}{3}}\) is considered an algebraic expression. Here, '27' is the number being operated on, the fractional exponent \(\frac{1}{3}\) indicates the operation of taking a cube root, and the result is a simpler expression - the number '3'. These algebraic expressions can become very complex, but it's important to remember they follow the same fundamental principles no matter how complicated they appear.
In the context of cube roots and radicals, the expression \(27^{\frac{1}{3}}\) is considered an algebraic expression. Here, '27' is the number being operated on, the fractional exponent \(\frac{1}{3}\) indicates the operation of taking a cube root, and the result is a simpler expression - the number '3'. These algebraic expressions can become very complex, but it's important to remember they follow the same fundamental principles no matter how complicated they appear.
Other exercises in this chapter
Problem 86
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