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Chapter 4: Problem 56

Write each expression using one exponent. $$5 \cdot 5$$

Short Answer

Expert verified
The expression \(5 \cdot 5\) written with exponents is \(5^2\).

Step by step solution

01

Understand the Basis of Exponentiation

An exponent refers to the number of times a number, or base, is multiplied by itself. For instance, \(5^3\) means \(5 \cdot 5 \cdot 5\). So we need to write the expression \(5 \cdot 5\) in an equivalent form that uses exponentiation.
02

Apply the Exponent

In the expression \(5 \cdot 5\), the number 5 is being multiplied by itself once. Thus, it can be rewritten as \(5^2\), where 2 represents the number of times 5 is being multiplied.

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Key Concepts

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

Exponentiation
Exponentiation is a mathematical operation that represents repeated multiplication of the same number. When you see an expression like \( a^n \), it means you multiply the number \( a \) by itself \( n \) times. For example, \( 5^3 \) means \( 5 \times 5 \times 5 \). It's a way to express large multiplication processes in a more compact form.

Here are a few points to help you understand exponentiation better:
  • The base is the number that is being multiplied.
  • The exponent is the number of times the base is multiplied by itself.
  • An exponent of 1 means the number stands alone: \( 5^1 = 5 \).
  • An exponent of 0 means the result is 1, regardless of the base: \( 5^0 = 1 \).
These insights help in simplifying expressions and are fundamental when dealing with powers in algebra.
Multiplication
Multiplication is one of the basic operations in mathematics that helps us find the total of repeated additions. In the context of exponentiation, it holds a specific significance where it ties directly into how powers are calculated.

Consider how multiplication works:
  • It takes two numbers, a "multiplier" and a "multiplicand", and calculates their product.
  • For example, in \( 5 \times 5 \), both 5 are multiplying each other, giving the product 25.
  • Through multiplication, you can break down or simplify expressions into more manageable forms.
For exponentiation, when you have \( 5 \cdot 5 \), you leverage multiplication twice with the same number. This is essentially saying \( 5^2 \), where multiplication shows its vital role in forming and understanding exponential terms.
Algebraic Expressions
Algebraic expressions are mathematical statements that include numbers, variables, and operations like addition, subtraction, multiplication, and division. In algebra, powers often simplify and solve expressions, making exponentiation a crucial component.

Explore the makeup of algebraic expressions:
  • An algebraic expression can be as straightforward as a single number or variable or complex with multiple operations.
  • When expressions have the same base being multiplied, like \( 5 \cdot 5 \), they can be simplified by using exponents.
  • Writing expressions with exponents not only reduces complexity but also makes calculations easier and faster.
In algebra, expressing \( 5 \cdot 5 \) as \( 5^2 \) is a great illustration of how exponentiation can tidy up mathematical expressions, paving the way for solving intricate problems efficiently.

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Most popular questions from this chapter

Writing the expression \((x+y)^{2}\) as \(x^{2}+y^{2}\) illustrates a common error. Explain.

How do you add like terms?

Perform the operations. $$-4\left(x^{2} y^{2}+x y^{3}+x y^{2} z\right)-2\left(x^{2} y^{2}-4 x y^{2} z\right)-2\left(8 x y^{3}-y\right)$$

Some students threw balloons filled with water from a dormitory window. The height \(h\) (in feet) of the balloons \(t\) seconds after being thrown is given by the polynomial function $$h=f(t)=-16 t^{2}+12 t+20$$ How far above the ground is a balloon 1.5 seconds after being thrown?

Perform the operations. $$\begin{aligned}&25 x^{3}-45 x^{2} z+31 x z^{2}\\\&-12 x^{3}+27 x^{2} z-17 x z^{2}\end{aligned}$$
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