Problem 20 For Exercises, simplify. $$\fr... [FREE SOLUTION]

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Chapter 9: Problem 20

For Exercises, simplify. $$\frac{w^{9}}{w}$$

Short Answer

Expert verified

$w^{8}$

Step by step solution

Identify the Given Element

The given expression to simplify is $\frac{w^{9}}{w}$.

Apply the Rule of Exponents

For power of the same base, the rule states that when you divide, you subtract the exponents. Therefore, $\frac{w^{9}}{w} = w^{(9-1)} = w^{8}$.

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Key Concepts

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

Exponents

Exponents are a way to express repeated multiplication of a number by itself. For example, any base, like $w$, raised to an exponent, $9$, means $w$ is multiplied by itself 9 times: $w \times w \times \dots \times w$ (9 times). It simplifies complex multiplications into a more manageable form. In using exponents:

$w^0$: Any number to the power of zero is 1, except zero.
$w^1$: Any number raised to the power of one is itself.
$w^n$: Indicates the base $w$ is multiplied by itself $n$ times.

Exponents follow precise rules that simplify expressions, especially when you deal with expressions having the same base.

Simplification

Simplification is the process of reducing an expression to its simplest form. This involves combining like terms and making the expression as concise as possible. The original expression given as $\frac{w^9}{w}$ is simplified by applying the rules of exponents. The goal here is to make the expression easier to understand and work with.
In simplifying algebraic expressions:

Identify like terms to combine them.
Use the rules of exponents to reduce terms.
Check if the final expression can be simplified further.

In this exercise, simplifying $\frac{w^9}{w}$ results in $w^8$, which is much simpler than the original form.

Division of Powers

The division of powers rule is central in simplifying expressions with exponents. When you divide terms with the same base, you subtract the exponent of the denominator from the exponent of the numerator. This rule is expressed as follows:
If you have $\frac{w^a}{w^b}$, then you simplify it to $w^{a-b}$. This means you subtract $b$ from $a$, as seen in this task where $\frac{w^9}{w}$ simplifies to $w^{9-1} = w^8$.
Key points in using the division of powers include:

Ensure that the bases are identical.
Subtract the exponent in the denominator from the exponent in the numerator properly.
Be cautious with negative exponents: they denote reciprocals.

Understanding this rule helps in efficiently expressing any divided terms with the same base in a more concise form.

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91影视

For Exercises, simplify. $$\frac{w^{9}}{w}$$

Short Answer

Step by step solution

Identify the Given Element

Apply the Rule of Exponents

Key Concepts

Exponents

Simplification

Division of Powers

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