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Chapter 4: Problem 4

Simplify where possible. Express your answer with positive exponents. a. \(\left(3 \cdot 3^{8}\right)^{-2}\) b. \(x^{3} \cdot x^{-4} \cdot x^{12}\) c. \(2^{6}+2^{6}+2^{7}+2^{-4}\) d. \(2 x^{-3}+3 x\left(x^{-4}\right)\) e. \(10^{-5}+5^{-2}+10^{10}\)

Short Answer

Expert verified
a. \frac{1}{3^{18}} \ b. x^{11} \ c. 2^{8} + 2^{-4} \ d. \frac{5}{x^{3}} \ e. 10^{-5} + 5^{-2} + 10^{10}

Step by step solution

01

Simplify \((3 \cdot 3^{8})^{-2}\)

First, simplify inside the parentheses. The expression inside the parentheses is:\[ 3 \cdot 3^{8} = 3^{1} \cdot 3^{8} = 3^{9} \]. Then apply the power rule \((a^{m})^{n} = a^{m \cdot n} \): \((3^{9})^{-2} = 3^{9 \cdot (-2)} = 3^{-18} \). To express the answer with positive exponents, rewrite it as: \((3^{-18}) = \frac{1}{3^{18}} \).
02

Simplify \(x^{3} \cdot x^{-4} \cdot x^{12}\)

Combine the exponents using the rule \(a^{m} \cdot a^{n} = a^{m+n}\). Thus: \((x^{3} \cdot x^{-4} \cdot x^{12}) = x^{3+(-4)+12} = x^{11}\).
03

Simplify \(2^{6} + 2^{6} + 2^{7} + 2^{-4}\)

First, combine like terms: \(2^{6} + 2^{6} = 2 \cdot 2^{6} = 2^{7}\). So, the expression reduces to \(2^{7} + 2^{7} + 2^{-4}\). Combine the first two terms: \(2^{7} + 2^{7} = 2 \cdot 2^{7} = 2^{8}\). Now, the expression is \(2^{8} + 2^{-4}\), which cannot be further simplified.
04

Simplify \(2 x^{-3} + 3 x (x^{-4})\)

Distribute in the second term: \(3 x (x^{-4}) = 3 x^{1} \cdot x^{-4} = 3 x^{1-4} = 3 x^{-3}\). Now combine like terms: \(2 x^{-3} + 3 x^{-3} = (2 + 3)x^{-3} = 5 x^{-3}\). Rewrite with positive exponents: \(5 x^{-3} = \frac{5}{x^{3}} \).
05

Simplify \(10^{-5} + 5^{-2} + 10^{10}\)

Each term is already in its simplest form and contains different bases, so the expression \(10^{-5} + 5^{-2} + 10^{10}\) cannot be simplified further.

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Key Concepts

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

Exponent Rules
Exponent rules are essential for simplifying expressions involving numbers and variables raised to powers. These rules help us combine and manipulate exponents systematically. There are several key rules:
  • Product Rule: When multiplying like bases, you add the exponents: \(a^m \times a^n = a^{m+n}\).
  • Power Rule: When raising a power to another power, you multiply the exponents: \((a^m)^n = a^{m \times n}\).
  • Quotient Rule: When dividing like bases, you subtract the exponent of the denominator from the numerator: \(\frac{a^m}{a^n} = a^{m-n}\).
  • Negative Exponent Rule: A negative exponent means taking the reciprocal of the base and changing the sign of the exponent: \(a^{-n} = \frac{1}{a^n}\).
  • Zero Exponent Rule: Any non-zero base raised to the power of zero is 1: \(a^0 = 1\).
Being familiar with these rules simplifies the process of handling complex algebraic expressions.
Algebraic Expressions
An algebraic expression is a combination of variables, numbers, and at least one arithmetic operation. Simplifying algebraic expressions often involves reducing them to their simplest form.
This process includes:
  • Applying exponent rules.
  • Combining like terms.
  • Distributing and factoring as needed.
  • Replacing negative exponents with positive ones.
For example, consider the expression: \((3 \times 3^8)^{-2}\). First, simplify inside the parentheses by using the product rule of exponents: \(3 \times 3^8 = 3^9\). Then, apply the power rule: \((3^9)^{-2} = 3^{-18}\). Finally, to express the answer with positive exponents, rewrite it as \(\frac{1}{3^{18}}\).
Understanding how to simplify such expressions is key to mastering algebra.
Positive Exponents
In algebra, it is often preferred to express answers with positive exponents. A positive exponent indicates the number of times a base is multiplied by itself.

For instance, when simplifying \(x^3 \times x^{-4} \times x^{12}\), use the product rule to combine exponents: \(3 + (-4) + 12 = 11\). The simplified expression is \(x^{11}\), which already has a positive exponent.

When dealing with negative exponents, convert them to positive by taking the reciprocal of the base. For example, if you have \(5x^{-3}\), rewrite it as \(\frac{5}{x^3}\). This keeps the expression more standard and often easier to interpret or further manipulate.
Combining Like Terms
Combining like terms is a foundational skill in algebra. Like terms have the same variables raised to the same power. You combine them to simplify expressions:

  • Firstly, identify like terms. For example, in \(2x^{-3} + 3x(x^{-4})\), the terms involving \(x^{-3}\) are like terms.

  • Next, distribute and simplify any products: \(3x(x^{-4}) = 3x^1 \times x^{-4} = 3x^{-3}\).

  • Then, combine the coefficients of these like terms: \(2x^{-3} + 3x^{-3} = 5x^{-3}\).
The final step is to express the answer with positive exponents if requested: \(5x^{-3} = \frac{5}{x^3}\).

Knowing how to identify and combine like terms simplifies algebraic expressions and is key in solving equations efficiently.

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