A combined operation example of a power of 2

Powers of Can combined, under the laws of exponents, to-create other Powers of. Under these rules, you can multiply powers of, divide powers of, or raise a power of both to a power and still get Another power of. You can combine these rules to create complicated expressions, expressions this result in a single power of. For example,

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The laws of exponents apply generally to any base; is no different. But since we ' re interested in powers of, we'll couch them in terms of powers of. Once we explain the laws in this-the-the-the-the-understand, the math behind the example above.

Basic Powers of the Rules

The following is the basic rules of exponents for powers of both, the rules you should is familiar with if you know what the P Owers of:

Rules for combining Powers of

The following is the rules for combining powers of the under multiplication, division, and exponentiation:

Product of Powersof both Rule:

This rule says a power of the multiplied by a powerof both is a power of. In particular, the resulting power of, a, and a exponent equal to the sum of the exponents of the multiplied of TW O.

Examples:

quotient of Powersof the Rule:

This rule says a power of the divided by a powerof both is a power of. In particular, the resulting power of both has a exponent equal to the dividend ' s exponent minus.

Examples:

power of a powerof the Rule:

This rule says a powerof the raised to a power is a power of. In particular, the resulting power of the "a" has a exponent equal to the product of the-the-the-given exponents.

Examples:

Composite Rules for combining Powers of

In the general laws of exponents there is the additional rules:the power of a product rule and the power of A quotient rule. For powers of, these rules become unnecessary, as they can is derived from the rules above. Nonetheless, they is still valid, and can provide a shortcut in calculations if your ' re so inclined.

Power of a Product of Powersof the Rule:

This rule is just the combination of the product of powers of both rule and the power of a power of both rule .

Examples:

Power of a quotient of Powersof the Rule:

This rule is just the combination of the quotient of the powers of the rule and the power of a power of both rule< /c5>.

Examples:

Powers of arbitrary Expressions

You can use the above rules-to-reduce complicated expressions to the form. Let's return to the example in the beginning of this article:

There is several ways to reduce this. Here is a-a-without using any composite (shortcut) Rules:

Here's a-on-the-using a composite rule:

Here's a longer, which would has been longer without the use of a composite rule:

Transforming Numeric Powers of

If there is powers of in a expression that appear numerically-for example, and instead of-transform them to the F Orm. Then, apply the rules above. Here are an example:

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A Visual Example

You can see the laws of exponents in action in the following diagram:

A 4 "x 4" x 4 "cube.

Here is three equivalent ways to compute the cube:

• Length x Width x Height: 4 x 4 x 4 = cubic inches
• Length x Width x Height: x x cubic inches
• Length cubed: cubic inches

Here is three equivalent ways to compute the area of each side of the cube:

• Length x Width: 4 x 4 = square inches
• Length x Width: square inches
• volume/height: square inches

Exercises

Answers

A combined operation example of a power of 2