Problem 42 Write with a single exponent. ... [FREE SOLUTION]

Chapter 6: Problem 42

Write with a single exponent. $$ 16^{2} y^{8} $$

Short Answer

Expert verified

Question: Rewrite the expression 16虏y鈦� with a single exponent. Answer: The expression 16虏y鈦� can be rewritten as 2^8y^8.

Step by step solution

Rewrite 16 using base 2

Since 16 is a power of 2 (16 = 2^4), we can rewrite the given expression as: $$ (2^4)^2 y^8 $$

Use properties of exponent to simplify

Using the property of exponents $(a^m)^n = a^{mn}$, we have: $$ (2^4)^2 y^8 = 2^{(4 \times 2)} y^8 $$

Combine exponents

The expression can now be written with a single exponent: $$ 2^{(4 \times 2)} y^8 = 2^8 y^8 $$

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Key Concepts

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

Properties of Exponents

When working with exponents, using their properties can simplify expressions significantly. One of the fundamental properties is the power of a power rule. This rule states that

$(a^m)^n = a^{m \cdot n}$

This means when you raise an exponent to another exponent, you multiply the powers together.
This property helps simplify complex expressions, where instead of calculating large numbers, you just handle the exponents.
Understanding these properties makes it easy to manage and reduce large expressions to a more straightforward form. It is this property that allows expressions like $(2^4)^2$ to become $2^8$.

Rewriting Expressions

Rewriting expressions is an essential skill, especially when simplifying or solving algebraic problems. By rewriting, you can unearth hidden relationships or simplify calculations. In our example, we began with $16^2$.
Recognizing that 16 is a power of 2 (specifically $2^4$), we rewrite the original expression. This transforms it into $(2^4)^2 y^8$, enabling us to use exponent properties to simplify further.
Rewriting doesn't change the value of an expression; it just recasts it in a form that can be easier to work with or understand.

Powers of Integers

Integer powers are simply repeated multiplication of an integer by itself. For instance, $16 = 2^4$ shows 16 as a power of 2 since multiplying four 2s together gives you 16:

$2 \times 2 \times 2 \times 2 = 16$

Recognizing numbers expressed in terms of powers can greatly simplify calculations, especially in algebraic contexts.
In our exercise, expressing 16 as a power of 2 enabled us to use exponent properties easily to simplify and solve the problem.
Working with powers often involves identifying simpler bases that can act as building blocks for larger numbers, simplifying both multiplication and division.

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91影视

Write with a single exponent. $$ 16^{2} y^{8} $$

Short Answer

Step by step solution

Rewrite 16 using base 2

Use properties of exponent to simplify

Combine exponents

Key Concepts

Properties of Exponents

Rewriting Expressions

Powers of Integers

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