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Chapter 0: Problem 43

Find each product. $$\left(5 r+3 t^{2}\right)^{2}$$

Short Answer

Expert verified
The expanded product is \(25r^2 + 30rt^2 + 9t^4\).

Step by step solution

01

Recognize the Formula

The expression \((5r + 3t^2)^2\) is a binomial raised to the power of 2. This can be expanded using the binomial square formula: \((a + b)^2 = a^2 + 2ab + b^2\), where \(a = 5r\) and \(b = 3t^2\).
02

Calculate \(a^2\)

Square the first term: \((5r)^2 = 25r^2\).
03

Calculate \(b^2\)

Square the second term: \((3t^2)^2 = 9t^4\).
04

Calculate \(2ab\)

Calculate two times the product of the first and second terms: \(2 \cdot 5r \cdot 3t^2 = 30rt^2\).
05

Combine All Parts

Use the expanded terms to express the binomial: \(25r^2 + 30rt^2 + 9t^4\). This is the expanded form of the expression.

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Key Concepts

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

Polynomials
A polynomial is an expression made up of variables, coefficients, and exponents, all combined using addition, subtraction, and multiplication. Polynomials can have one or more terms. Each term in a polynomial consists of:
  • A variable (such as \(r\) or \(t\) in our example) that can take different values.
  • An exponent that denotes the power to which the variable is raised (2 in \(r^2\)).
  • A coefficient, which is a constant multiplied by the variable (like 5 in \(5r\)).
Polynomials are classified based on their degree and the number of terms. The degree of a polynomial is determined by the highest exponent it contains. For instance, in the polynomial \(25r^2 + 30rt^2 + 9t^4\), the degree is 4, contributed by the term \(9t^4\). Understanding polynomials is key in algebra as they form the foundation for many mathematical concepts.
Algebraic Expressions
Algebraic expressions include numbers, variables, and operations, all arranged to describe a specific quantity or formula. In simple terms, they represent a statement of equality or an expression of calculation using algebraic symbols.

An example of such an expression is \((5r + 3t^2)^2\), which involves both addition and exponentiation. The purpose of algebraic expressions is to succinctly convey mathematical ideas, relationships, or structures.
  • Terms like \(5r\) and \(3t^2\) are algebraic terms that can be combined or simplified depending on the operations involved.
  • The expression can be evaluated for particular values of the variables \(r\) and \(t\), providing numeric results.
  • They can represent equations or inequalities and are pivotal in forming models in science and engineering.
Understanding how to manipulate and simplify algebraic expressions is a crucial skill in solving complex equations.
Exponentiation
Exponentiation is a mathematical operation involving two numbers: the base and the exponent. It simplifies repeated multiplication of the same number or expression.

For example, in the expression \((5r + 3t^2)^2\), the 2 is the exponent, applied to the base \((5r + 3t^2)\). This means to multiply \((5r + 3t^2)\) by itself:
  • This operation uses the power of 2, simplifying repeated multiplication by indicating it concisely.
  • The binomial expansion using exponentiation is crucial, applying the formula \((a + b)^2 = a^2 + 2ab + b^2\), an example of exponentiation in action.
  • Exponentiation in algebra is just one way it shows up; it's also prevalent in areas like calculus, physics, and engineering.
The concept of exponentiation is important in algebraic operations as it plays a significant role in calculations and can simplify otherwise complex mathematical expressions.

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Most popular questions from this chapter

Write each expression in radical form. Assume that all variables represent positive real numbers. $$\sqrt[5]{k^{2}}$$

Simplify each expression, assuming that all variables represent nonnegative real numbers. $$\sqrt[3]{32}-5 \sqrt[3]{4}+2 \sqrt[3]{108}$$

Simplify each expression, assuming that all variables represent nonnegative real numbers. $$\frac{-4}{\sqrt[3]{3}}+\frac{1}{\sqrt[3]{24}}-\frac{2}{\sqrt[3]{81}}$$

Rationalize the denominator of each radical expression. Assume that all variables represent nonnegative real numbers and that no denominators are \(0 .\) $$\frac{\sqrt{7}-1}{2 \sqrt{7}+4 \sqrt{2}}$$

Find each product. Assume that all variables represent positive real numbers. $$\left(2 z^{1 / 2}+z\right)\left(z^{1 / 2}-z\right)$$
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