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Chapter 2: Problem 22

Identify the base and the exponent in each. $$\frac{5}{9} t^{7}$$

Short Answer

Expert verified
In the expression \(\frac{5}{9} t^7\), the base is \(t\) and the exponent is \(7\).

Step by step solution

01

Recognize the base

In the expression \(\frac{5}{9} t^7\), the base is the number or variable that is being raised to a power. In this case, the base is the variable \(t\).
02

Recognize the exponent

In the expression \(\frac{5}{9} t^7\), the exponent is the power to which the base is raised. In this case, the exponent is \(7\). So, the base is \(t\) and the exponent is \(7\).

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Key Concepts

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

Algebraic Expressions
Algebraic expressions are a cornerstone of algebra. They are combinations of numbers, variables, and mathematical operations. An algebraic expression can be as simple as a single number or variable, or more complex with multiple terms and operations. For instance, in the expression \(\frac{5}{9}t^7\), each component plays a distinct role:
  • \(\frac{5}{9}\) serves as a constant coefficient. It is the numerical part of the expression that multiplies the variable.
  • \(t^7\) includes both the variable \(t\) and an exponent 7.

Understanding algebraic expressions helps in solving equations, simplifying expressions, and modeling real-world scenarios. Recognizing the role of coefficients and variables within expressions is crucial for a deeper comprehension of algebra.
Variables in Algebra
Variables are an integral part of algebra. They are symbols that represent unknown values or quantities. Typically, letters such as \(x\), \(y\), or \(t\) are used as variables. In the expression \(\frac{5}{9}t^7\), t is the variable, serving as the placeholder for a number we do not yet know.

A variable can take on different values, and that's what makes algebra flexible and powerful. By manipulating variables, you can solve for unknowns, find patterns, and generalize mathematical relationships. Understanding how to manage variables helps in constructing and interpreting mathematical models.
Powers and Exponents
Powers and exponents are ways to express repeated multiplication compactly. In an expression like \(t^7\), the number \(7\) is the exponent, and \(t\) is the base.
  • The base \(t\) indicates what number or variable you are multiplying by itself.
  • The exponent \(7\) shows how many times you use the base in a multiplication. That is, \(t^7\) means \(t \times t \times t \times t \times t \times t \times t\).

Using exponents makes complex arithmetic easier to write and understand. They follow specific rules and laws, such as the product rule and power rule, which simplify calculations involving powers. Mastery of powers and exponents opens up new possibilities in both algebraic and geometrical problem-solving.

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Most popular questions from this chapter

Simplify the expression using the product rule. Leave your answer in exponential form. $$\left(-7 t^{6}\right)\left(t^{3}\right)\left(-4 t^{7}\right)$$

Express each number in scientific notation, then solve the problem. Humans shed about \(1.44 \times 10^{7}\) particles of skin every day. How many particles would be shed in a year? (Assume 365 days in a year.)

Simplify using the quotient rule. $$\frac{200 x^{x} y^{3}}{20 x^{10} y^{11}}$$

Perform the operation as indicated. Write the final answer without an exponent. \(\frac{8 \times 10^{7}}{4 \times 10^{4}}\)

Write each number in scientific notation. The diameter of an atom is about \(0.00000001 \mathrm{cm}\)
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