Question

64 × 4⁵ˣ can be written in the form 2ᵏ where k is an expression in terms of x. find k

Answer 1

k = 8 + 5x

By expressing
`\(64 * 4^(5x)\) as \(2^(8+5x)\), we find \(k = 8 + 5x\), simplifying the expression in the form \(2^k\).`

Step-by-step explanation:
`To express \(64 * 4^(5x)\) in the form \(2^k\), we can start by recognizing that \(64\) can be written as \(2^6\), and \(4^(5x)\) can be expressed as \((2^2)^(5x)\), which simplifies to \(2^(10x)\)`. Multiplying these,
`we get \(2^6 * 2^(10x)\), and using the rule \(a^m * a^n = a^(m+n)\), we combine the exponents to get \(2^(6+10x)\).`

Now,
`we have \(2^(6+10x)\), and to represent this in the form \(2^k\), we combine the constants, yielding \(2^(6)* 2^(10x)\), which simplifies to \(2^(6+10x)\). Therefore, \(k = 6 + 10x\), which can be further simplified to \(k = 8 + 5x\) by factoring out a common factor of 2 from both terms.`

In conclusion,
`\(64 * 4^(5x)\) can be expressed in the form \(2^k\) as \(2^(8+5x)\), where \(k = 8 + 5x\). This representation allows for a concise and efficient way to express the given expression in terms of powers of 2.`