Problem 35 - Algebra 1

Question

Simplify the expression. The simplified expression should have no negative exponents. $$ x^{5} \cdot \frac{1}{x^{8}} $$

Step-by-Step Solution

Answer

The simplified expression with no negative exponents for \(x^{5} * 1/x^{8}\) is \(1 / x^{3}\).

Step 1: Recognize the division of exponents

The given expression is a division of two numbers with exponents. When you divide two numbers with the same base, the rule of exponents states that you subtract the exponent of the denominator from that of the numerator. In this case, both numbers have the base of \(x\), so you can apply the rule: \(x^{n} / x^{m} = x^{(n-m)}\). Thus the given expression \(x^{5} * 1/x^{8}\) simplifies to \(x^{(5-8)}\).

Step 2: Simplify the exponent

Now simplifying the exponent inside the brackets: \(x^{(5-8)} = x^{-3}\). This results in a negative exponent.

Step 3: Convert to positive exponent

As per the rules of exponents, a negative exponent can be converted to a positive exponent by taking its reciprocal. Thus, \(x^{-3}\) can be written as \(1 / x^{3}\).

Key Concepts

Simplifying Expressions

The concept of simplifying expressions involves reducing them into their simplest form, making calculations easier and more efficient. In mathematics, simplifying is akin to tidying up a room – you remove unnecessary elements and keep everything organized.

When simplifying expressions with exponents, you should follow specific rules to combine like terms and exponents. For instance, when multiplying expressions with the same base, you add the exponents. Conversely, when dividing them, you subtract the exponents. In the original exercise, for example, the expression involves both multiplication and division of the same base \(x\). The simplification rule applied is:

• For multiplication: \(x^{a} \cdot x^{b} = x^{(a+b)}\)
• For division: \(x^{a} \cdot \frac{1}{x^{b}} = x^{(a-b)}\)

In the case of \(x^{5} \cdot \frac{1}{x^{8}}\), simplifying results in \(x^{(5-8)} = x^{-3}\). Understanding this process is essential as it helps reduce complex expressions into manageable parts, which is crucial in algebra.

Negative Exponents

Negative exponents might seem challenging at first, but they follow straightforward rules. A negative exponent denotes the reciprocal of the base raised to the positive power of that exponent.

For example, \(x^{-n}\) means \(1 / x^{n}\). The negative sign flips the base to its reciprocal while converting the exponent to a positive number. This is a way of expressing how many times the reciprocal of a number is used in a multiplication.

Applying this to our simplified expression \(x^{-3}\), it means we take the reciprocal of \(x\) three times yielding \(1 / x^{3}\).

Key things to remember about negative exponents include:

• Negative exponents move the base across the fraction line, turning it into a positive exponent.
• They give the same result as multiplying one by the fraction of the base, raised to the corresponding positive power.

Learning to manipulate negative exponents is pivotal, as it enables you to handle and simplify complex expressions effectively in mathematics.

Reciprocal of Exponents

The reciprocal of exponents is a concept closely tied to negative exponents. When a base raised to a negative exponent is inverted, the result is said to be its reciprocal.

The idea of a reciprocal is fundamental in mathematics. For any nonzero number \(a\), its reciprocal is \(1/a\). Similarly, when dealing with exponents, the reciprocal of \(x^{-n}\) is \(x^{n}\), because \(x^{-n} = 1/x^{n}\).

In our specific problem where we ended with \(x^{-3}\), we convert it to its reciprocal form, obtaining \(1/x^{3}\). This transformation simplifies and clarifies the expression, eliminating negative exponents.

Some key points regarding the reciprocals of exponents include:

• Reciprocal expressions make problem-solving easier by converting negative exponents to positive ones.
• This conversion is essential for expressing an equation or expression in its simplest, most interpretable form.
• It is extensively used in equations, calculus, and more complex mathematical calculations.

By mastering the reciprocal transformation, you'll find simplifying expressions with negative exponents much more manageable.