Problem 14
Question
Fill in the blanks. \(\sqrt[3]{x}=y \text { if } y^{3}=\)
Step-by-Step Solution
Verified Answer
\( x \)
1Step 1: Understanding the Problem
When we see the expression \( \sqrt[3]{x} = y \), we're dealing with a cube root. This expression states that \( y \) is the number which, when cubed, equals \( x \). Therefore, filling in the blanks requires expressing this relationship in an equation form.
2Step 2: Formulating the Equation
To find what expression equals \( y^3 \), we reformulate \( \sqrt[3]{x} = y \) into an equation involving \( y^3 \). Since \( y \) is the cube root of \( x \), when we cube \( y \), we should get \( x \). Thus, \( y^3 = x \).
Key Concepts
AlgebraEquation SolvingExponents
Algebra
Algebra is a language of mathematics that allows us to express mathematical relationships and problem solve using symbols and letters to represent numbers and operations. At its core, algebra involves manipulating these symbols to find unknown values.
One of the fundamental concepts in algebra is the use of equations to represent relationships. In our exercise, the expression \( \sqrt[3]{x} = y \) indicates a specific relationship between \( x \) and \( y \), using a cube root. This requires understanding how to transition between the operation of taking cube roots and performing operations with exponents.
Using algebraic manipulation, we transform expressions like \( \sqrt[3]{x} \) into equations like \( y^3 = x \). This is important because it shows how a potentially complex term can be rewritten in a form that is often easier to work with.
One of the fundamental concepts in algebra is the use of equations to represent relationships. In our exercise, the expression \( \sqrt[3]{x} = y \) indicates a specific relationship between \( x \) and \( y \), using a cube root. This requires understanding how to transition between the operation of taking cube roots and performing operations with exponents.
Using algebraic manipulation, we transform expressions like \( \sqrt[3]{x} \) into equations like \( y^3 = x \). This is important because it shows how a potentially complex term can be rewritten in a form that is often easier to work with.
- Algebra simplifies problem-solving by using variables to represent unknown quantities.
- Equations are used to model real-world situations.
- Cube roots can be denoted as exponents, which helps to simplify algebraic expressions.
Equation Solving
Equation solving is a process used to find unknown numbers or quantities represented by variables. It involves moving parts of an equation around using algebraic operations.
When we solve an equation like \( \sqrt[3]{x} = y \), we are essentially seeking to understand the equality and balance between the expressions on either side of the equation. In this specific scenario, solving for \( y \) requires recognizing that cube-rooting \( x \) yields \( y \), and taking \( y \) to the power of 3 yields \( x \).
Understanding and solving such equations helps us:
When we solve an equation like \( \sqrt[3]{x} = y \), we are essentially seeking to understand the equality and balance between the expressions on either side of the equation. In this specific scenario, solving for \( y \) requires recognizing that cube-rooting \( x \) yields \( y \), and taking \( y \) to the power of 3 yields \( x \).
Understanding and solving such equations helps us:
- Identify relationships between mathematical expressions.
- Determine unknown values through systematic manipulation.
- Learn how reverse operations work, such as understanding cube roots through exponentiation.
Exponents
Exponents are a powerful mathematical tool that involve raising numbers to a power. This means multiplying a number by itself a certain number of times. In our exercise, understanding that \( y^3 = x \) allows us to comprehend the concept of a cube, which is an example of an exponent frequently used in mathematics.
An exponent of 3 indicates that a number is multiplied by itself two more times after the first instance, such as \( y \times y \times y \). Conversely, taking the cube root \( \sqrt[3]{x} \) is the action of determining what number \( y \) must be, so that when it is multiplied by itself 3 times, it results in \( x \).
An exponent of 3 indicates that a number is multiplied by itself two more times after the first instance, such as \( y \times y \times y \). Conversely, taking the cube root \( \sqrt[3]{x} \) is the action of determining what number \( y \) must be, so that when it is multiplied by itself 3 times, it results in \( x \).
- Exponents signify repeated multiplication.
- Cube roots and exponents are inverse operations.
- Exponents simplify calculations involving large numbers.
Other exercises in this chapter
Problem 14
Complete each rule for exponents. $$ \frac{1}{x^{-m / n}}= \square $$
View solution Problem 14
Complete each solution. $$ \begin{aligned} \frac{3}{2-i} &=\frac{3}{2-i} \cdot \\ &=\frac{6+}{4-} \\ &=\frac{6+3 i}{5} i \\ &=-+\frac{3}{5} \end{aligned} $$
View solution Problem 14
Simplify each expression. $$ \sqrt{28} $$
View solution Problem 14
Solve \(8^{2}+4^{2}=c^{2}\) and assume \(c>0 .\) Approximate to two decimal places. (EQUATION CAN'T COPY)
View solution