Question

Write each expression with base 2 a) \(4^{6}\) b) \(8^{3}\) c) \(\left(\frac{1}{8}\right)^{2}\) d) 16

Short answer

a) 2^{12}, b) 2^9, c) 2^{-6}, d) 2^4

Step-by-step solution

Understand the Powers of 2

Recognize that numbers like 4, 8, 16 can be written as powers of 2. Specifically: - 4 can be written as 2^2 - 8 can be written as 2^3 - 16 can be written as 2^4

Convert 4 to Base 2 in Expression a)

Since 4 = 2^2, substitute 4 with 2^2 in the expression: \[4^6 = (2^2)^6 = 2^{12}\]

Convert 8 to Base 2 in Expression b)

Since 8 = 2^3, substitute 8 with 2^3 in the expression: \[8^3 = (2^3)^3 = 2^9\]

Convert 8 to Base 2 in Expression c)

Since 8 = 2^3, substitute 8 with 2^3 in the fraction: \[\frac{1}{8} = 2^{-3}\] Then raise the result to the 2nd power: \[\bigg(\frac{1}{8}\bigg)^2 = (2^{-3})^2 = 2^{-6}\]

Express 16 in Base 2

Since 16 = 2^4, it directly converts to: \[16 = 2^4\]

Key concepts

Powers of 2

To understand how to write expressions in base 2, it’s crucial to know the concept of 'powers of 2'. This means expressing numbers as a product of 2 raised to an exponent. For example:

• 4 can be written as \(2^2\)
• 8 can be written as \(2^3\)
• 16 can be written as \(2^4\)

Recognizing these patterns helps in converting numbers to base 2 more easily and is an essential skill for working with different bases in mathematics.

Exponential Expressions

An exponential expression involves raising a base number to a certain power. For instance, in the expression \(4^6\), 4 is the base and 6 is the exponent. Converting such expressions to base 2 involves a few steps.
First, identify how the base number can be rewritten as a power of 2. In the case of 4, since \(4 = 2^2\), the expression \(4^6\) can be transformed to \((2^2)^6\).
Using the laws of exponents, particularly the power of a power rule, simplify the expression to \(2^{12}\). This process is used for all given expressions to achieve their base 2 form.

Fractional Exponents

Sometimes, you need to work with fractional exponents or negative exponents, especially when dealing with fractions or roots. For example, consider the expression \(\bigg(\frac{1}{8}\bigg)^2\).
Notice that \(\frac{1}{8}\) can be written as \(8^{-1}\). Since 8 can also be written as \(2^3\), \(\frac{1}{8}\) becomes \(2^{-3}\).
Now, raising this to the power of 2, we get \((2^{-3})^2 = 2^{-6}\). Recognizing how to handle negative and fractional exponents allows you to convert a wider range of numbers to base 2 efficiently.

Understanding Mathematical Bases

A mathematical base refers to the number that gets multiplied by itself a certain number of times in an exponential expression. The most common base is 10, but base 2 is also frequently used, especially in computing and digital logic.
Converting numbers to different bases involves understanding how numbers can be decomposed into sums of powers of the new base. For example, 16 in base 2 can directly be written as \(2^4\).
This means that familiarity with different mathematical bases and how to express numbers in these bases is a crucial skill. Exercises that involve rewriting numbers in base 2 strengthen your ability to think in these different representations, making it easier to solve related problems in both academics and real-life applications like computer science.