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Chapter 5: Problem 5

Write each expression as a cube. $$y^{12}$$

Short Answer

Expert verified
The expression \(y^{12}\) as a cube is \((y^{4})^{3}\).

Step by step solution

01

Understand the Expression

We have the expression \(y^{12}\). According to the rules of exponents, the exponent tells us how many times the base, y, is used as a factor. However, the problem is asking us to express this as a cube.
02

Express as a Cube

A cube of a number means the number multiplied by itself twice, which can be represented as \(n^3\). Therefore, we want to find a way to express \(y^{12}\) as \((something)^3\), which will represent it as a cube. This can be done by observing that 12 divided by 3 is 4. Hence, we can rewrite \(y^{12}\) as \((y^4)^3\).
03

Final Expression

There is nothing more to simplify, so the final expression as a cube is \((y^4)^3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponents
Exponents are a fundamental concept in mathematics that dictate how many times a base number is multiplied by itself. For example, the expression \(y^{12}\) means the base \(y\) is multiplied by itself 12 times. Exponents are written as superscripts to the top right of the base number. This compact notation helps simplify the representation of large multiplications.
  • For instance, \(y^2\) means \(y \times y\).
  • \(y^3\) translates to \(y \times y \times y\).
  • So, \(y^{12}\) is \(y \times y \times \ldots \text{(12 times)}\).

When working with higher powers like \(y^{12}\), breaking them down into smaller components, such as cubes, can simplify calculations and improve understanding.
Cubic expressions
Cubic expressions involve raising a number or a variable to the power of three, represented as \(n^3\). This means you multiply the base three times. For example, the cube of \(y\) is \(y^3 = y \times y \times y\). Cubic expressions often show up in algebra, geometry, and various real-world applications, particularly when dealing with volumes, as they increase the dimensions of the base in threefold.
  • A cube has three equal sides, so mathematically and visually, cubing a number involves three dimensions.
  • Cubic numbers grow quickly as the base number increases.
  • For example, cubing 2 gives \(2^3 = 8\), while cubing 4 gives \(4^3 = 64\).

To express \(y^{12}\) as a cube, we can structure it such that the exponent is a multiple of three, e.g., \((y^4)^3\). This means \(y^{12} = (y^4 \times y^4 \times y^4)\).
Algebraic expressions
Algebraic expressions combine numbers, letters (variables), and operation signs to represent a value or relationship. In algebra, these expressions can be manipulated using various operations to simplify or solve equations. An expression like \(y^{12}\) falls under this category because it includes a variable and an operation (exponentiation).
  • Variables in algebra serve as placeholders for unknown or variable quantities.
  • Operations like addition, subtraction, multiplication, division, and exponentiation are used to form expressions.
  • Expressions can be combined or transformed using algebraic rules.

Writing \(y^{12}\) as \((y^4)^3\) is an example of transforming an algebraic expression. This transformation helps us see the expression in a different light, showcasing another structure that may be more useful for solving certain types of problems.

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Most popular questions from this chapter

Factor. Write each trinomial in descending powers of one variable, if necessary. If a polynomial is prime, so indicate. $$r^{2}-2 r s+4 s^{2}$$

Factor. If the polynomial is prime, so indicate. $$-12 y^{2}-12+25 y$$

Factor. If the polynomial is prime, so indicate. $$4 a^{2}-15 a b+9 b^{2}$$

Factor. $$m^{2}+8 m+16-n^{2}$$

Factor each expression completely. $$3 a b+9 a-2 b-6$$
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