Problem 37 - Algebra 2 and Trigonometry

Question

In \(35-63,\) write each expression with only positive exponents and express the answer in simplest form. The variables are not equal to zero. $$ y^{-5} $$

Step-by-Step Solution

Answer

The expression \( y^{-5} \) is simplified to \( \frac{1}{y^5} \).

Step 1: Understand Negative Exponents

Recall that a negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. For example, for any non-zero number \(a\) and integer \(n\), \(a^{-n} = \frac{1}{a^n}\).

Step 2: Apply the Negative Exponent Rule

The given expression is \(y^{-5}\). According to the rule for negative exponents, we rewrite it as \(\frac{1}{y^5}\). This transforms the negative exponent into a positive one by taking the reciprocal of the base.

Step 3: Simplify the Expression

Since the expression \(\frac{1}{y^5}\) is already in its simplest form with no further simplification possible, this is your final answer.

Key Concepts

Reciprocal

Negative exponents can seem a bit tricky at first, but once you understand the concept of reciprocal, it becomes simpler. When you see a negative exponent, think of it as an instruction to flip the base of the number. If you have a number such as \(a^{-n}\), it is equivalent to its reciprocal \(\frac{1}{a^n}\).

The term "reciprocal" means to take the inverse of a number. For example, the reciprocal of 3 is \(\frac{1}{3}\). When we apply this idea to exponents, a negative exponent tells us to "invert" the number it's applied to, turning \(a^{-n}\) into \(\frac{1}{a^n}\).

• This is how negative exponents shift from the numerator to the denominator or vice versa.
• It's crucial to remember that the base must be non-zero because division by zero is undefined.

This rule helps transform expressions with negative exponents into more familiar positive integer exponents, which are easier to work with.

Simplifying Expressions

Simplifying mathematical expressions involves reducing them to their simplest form while retaining their original value. When you simplify expressions with exponents, especially negative ones, you often start by rewriting them with only positive exponents.

Consider \(y^{-5}\). Simplifying it means applying the rule we discussed earlier: replacing the negative exponent with the reciprocal of the base raised to the positive exponent. Hence, \(y^{-5}\) becomes \(\frac{1}{y^5}\). This expression is now simplified because it meets the criterion of having all positive exponents.

• Always check for like terms or factors that may combine during simplification.
• Remember that simplification doesn't change the value of the expression, only its form.

Simplifying expressions lets you work with them more easily, especially during further calculations or evaluations.

Positive Exponents

Positive exponents are much more straightforward than their negative counterparts. They simply instruct you to multiply the base by itself as many times as the exponent specifies. For example, \(y^5\) means you multiply \(y\) by itself five times: \(y \times y \times y \times y \times y\).

Using positive exponents often makes expressions more manageable and visually clearer. This step is particularly important in algebra, where simplifying expressions to their basic components can significantly streamline problem-solving.

• Positive exponents denote repeated multiplication, which is foundational for understanding powers and roots.
• They are essential in polynomial expressions and equations.

When simplifying expressions, transitioning from negative to positive exponents not only aligns with standard algebraic practices but also prepares us for more complex mathematical manipulations.