Problem 12 - Elementary and Intermediate Algebra

Question

Fill in the blanks. a. We read \(n^{2} n^{3} n\) as "n _____ times \(n\) _____M times \(n\). b. We read \(\frac{x^{7}}{x^{5}}\) as " \(x\) to the seventh power ____ by \(x\) to the _____ power: c. We read \((b+5)^{6}(b+5)^{8}\) as "the _____ of \(b+5\) _____ to the sixth power, times the _____ of \(b+5\) raised to the _____ power."

Step-by-Step Solution

Answer

a. two, three; b. divided, fifth; c. binomial, raised, binomial, eighth.

Step 1: Understanding the Expression

In this exercise, we need to interpret algebraic expressions in terms of exponents and operations. This involves translating the expressions into words according to the patterns of multiplication (for same bases) and division (for quotients of powers).

Step 2: a: Expression Interpretation

The expression given is \(n^2 n^3 n\). Each \(n\) is being multiplied. When exponents with the same base are multiplied, their exponents are added. Thus, the entire expression simplifies to \(n^6\). The explanation is "\(n\) multiplied two times, \(n\) three times, and \(n\) one time."

Step 3: b: Fractional Power Interpretation

The expression \(\frac{x^7}{x^5}\) involves dividing powers with the same base. According to the rules of exponents, we subtract the exponent in the denominator from the exponent in the numerator. So, it simplifies to \(x^2\). We read it as "\(x\) to the seventh power divided by \(x\) to the fifth power."

Step 4: c: Binomial Multiplication Interpretation

For \((b+5)^6(b+5)^8\), we have two binomials with the same base, \((b+5)\), and different exponents multiplied. Their exponents are added: \((b+5)^{6+8} = (b+5)^{14}\). We describe this as "the binomial of \(b+5\) raised to the sixth power, times the binomial of \(b+5\) raised to the eighth power."

Key Concepts

Exponents

Exponents are a way of expressing repeated multiplication of a number by itself. They make it easier to read and write numbers. For example, instead of writing out a number like \( n \) multiplied by itself multiple times, we use exponents. The exponent tells us how many times to use the number in a multiplication. So, \( n^2 \) means \( n \) multiplied by itself two times: \( n \times n \). Similarly, \( n^3 \) would be \( n \) multiplied by itself three times: \( n \times n \times n \). In the expression \( n^2 n^3 n \), we see the combination of these exponents and the base \( n \). By adding the exponents, we simplify it to \( n^6 \), representing \( n \) multiplied by itself six times overall. Exponents simplify complex expressions and are a key concept in algebra.

Multiplication of Powers

When multiplying powers with the same base, we add their exponents. This is a crucial concept for simplifying expressions, and it applies when handling expressions like \( a^m \times a^n \). According to the rules of exponents, the expression becomes \( a^{m+n} \). Consider the multiplication \( n^2 \times n^3 \); here the base \( n \) remains the same, and the exponents are added: \( 2 + 3 \). This results in \( n^5 \). The same principle is applied in any situation where you multiply powers with the same base. The statement in the exercise where it says "\( n \) multiplied two times, \( n \) three times, and \( n \) one time," relies on this rule. That's how we simplify \( n^2 \times n^3 \) to \( n^6 \), as the additional \( n \) (\( n^1 \)) needs its exponent added as well.

Division of Powers

Division of powers simplifies expressions by subtracting the exponents when the bases are the same. This rule is a handy tool for simplifying fractions, commonly seen in algebraic manipulation. Consider an expression like \( \frac{x^a}{x^b} \). When dividing, you subtract the exponent in the denominator from the one in the numerator: \( x^{a-b} \). For example, the expression \( \frac{x^7}{x^5} \) involves subtracting 5 from 7, resulting in \( x^2 \). This simplification reduces complexity by standardizing the expression with the same base and fewer components. That's why the exercise reads it as "\( x \) to the seventh power divided by \( x \) to the fifth power." This mechanism helps maintain mathematical expressions in a concise form.

Binomial Expression

A binomial expression includes two terms that are combined, often expressed as \( (a+b) \). When we deal with powers of binomials, rules similar to single-term expressions come into play. In a situation like \((b+5)^6(b+5)^8\), each term is a binomial raised to a power and then multiplied. We apply the multiplication of powers rule: add the exponents when the bases are the same. Therefore, \((b+5)^6(b+5)^8\) becomes \((b+5)^{6+8}\) or \((b+5)^{14}\). This exemplifies manipulating binomial expressions with exponents for simplification. It's described as "the binomial of \( b+5 \) raised to the sixth power, times the binomial of \( b+5 \) raised to the eighth power." Understanding how to work with these expressions helps in algebraic operations like expansion and solving equations.