Problem 66 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ 2^{3} x^{2} 2^{-3} x^{-2} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $$2^3 x^2 2^{-3} x^{-2}$$ and write it with only positive exponents. Answer: 1

Step 1: Combine the terms with the same base, in this case 2

Using the properties of exponents, we combine the terms with the base 2: $$2^3 \cdot 2^{-3} = 2^{3 + (-3)}$$ The exponents add up to zero (3 + (-3) = 0). We know that $$a^0 = 1$$ for any nonzero a, so: $$2^{3 + (-3)} = 2^0 = 1$$ So our expression becomes: $$1 \cdot x^2 \cdot x^{-2}$$

Step 2: Combine the terms with the same base, in this case x

Now we can combine the terms with the base x: $$x^2 \cdot x^{-2} = x^{2 + (-2)}$$ The exponents add up to zero (2 + (-2) = 0), and since $$a^0 = 1$$: $$x^{2 + (-2)} = x^0 = 1$$ So our expression becomes: $$1 \cdot 1$$

Step 3: Simplify the expression

Since the product of 1 and 1 is 1, $$1 \cdot 1 = 1$$ So the simplified expression with only positive exponents is 1.

Key Concepts

Exponents and Powers

When we talk about exponents and powers, we are dealing with a shorthand notation for repeated multiplication. The exponent tells us how many times to multiply the base by itself. For example, in the expression \( 2^3 \), 2 is the base and 3 is the exponent, which implies \( 2 \times 2 \times 2 \).Positive exponents are straightforward: \( a^n \) means we multiply \( a \) by itself \( n \) times if \( n \) is a positive integer. It is important to understand that when the exponent is 0, as in \( a^0 \), the result is always 1, given \( a \) is not zero. This is because multiplying something zero times means we are not really multiplying at all, which leaves us with a multiplicative identity of 1.

Exponential Expressions

An exponential expression consists of a base raised to an exponent. It's a compact way to represent a number that could be extremely large or small—hence the term \( 'exponential' \).

For example, \( 2^{3} x^{2} \), contains two exponential expressions. They are meaningful in various branches of math, science, and engineering, where the range of quantities can be immense, and exponential notation keeps numbers manageable and legible. Understanding how to manipulate these expressions, including when working with negative exponents, helps in simplifying complex equations and is vital when progressing in mathematical studies.

Properties of Exponents

The properties of exponents are rules that describe how exponential expressions can be manipulated or combined. They make working with exponents much easier and enable us to simplify expressions without performing all of the multiplications.

• Addition of exponents (Product Rule): \( a^m \times a^n = a^{m+n} \) where \( a \) is the base and \( m \) and \( n \) are exponents.
• Zero exponent rule: \( a^0 = 1 \) for any nonzero \( a \).
• Negative exponent rule: \( a^{-n} = 1/a^n \) where \( a \) is nonzero. This tells us that negative exponents represent reciprocal values.

These properties allow us to transform expressions with negative exponents into equivalent expressions with positive exponents, simplifying the expressions and making them easier to work with.

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operational symbols. Expressions can be as simple as \( x+5 \) or as complex as \( 2^{x^2} - 4/x \) where variables such as \( x \) represent unknown values.In the exercise, \( 2^{3} x^{2} 2^{-3} x^{-2} \) is an algebraic expression where the variables and constants are expressed with exponents. When simplifying algebraic expressions with exponents, it's crucial to apply the properties of exponents correctly. In doing so, you can manipulate them, often simplifying the expression to a more understandable form or to prepare the expression for solving equations.