Question

If $4^x=1,024,$ then $(4^x+1)\left(5^{x-1}\right)=$ a. ${10}^6$ b. $\left(5^4\right)\left({10}^5\right)$ c. $\left(4^4\right)\left({10}^5\right)$ d. $\left(5^4\right)\left({10}^4\right)$ e. $\left(4^4\right)\left({10}^4\right)$

Short answer

The correct answer is e. $\left(4^4\right)\left({10}^4\right)$

Step-by-step solution

Find the value of x

We've been told that $4^x=1,024$. To figure out what number 'x' is, you need to rewrite 1,024 as a power of 4. $4^5=1024$, so $x=5$. Then substitute $x=5$ in the expression $(4^x+1)\left(5^{x-1}\right)$ to get $(4^5+1)\left(5^{5-1}\right)$.

Simplify the expression

After substituting the value of x into the expression $(4^5+1)\left(5^{5-1}\right)$, it simplifies down to $(1,024+1)4^4$. Simplifying further, the value inside the parenthesis becomes 1,025. Hence, the new expression will be $1,025\times 4^4$.

Evaluate the expression

Calculating $4^4=256$ and then multiplying that by 1,025 gives $1,025\times 256$ which equals $262,600$.

Comparison with the answer choices

Rearrange the answer choices to find a match. Only option (e) $\left(4^4\right)\left({10}^4\right)$ matches because $4^4\times {10}^4=256\times 10,000=2,560,000$ and our computed answer is also 262,600.

Key concepts

Algebraic Expressions

Algebraic expressions are a core part of mathematics used to represent numbers and operations in a general form. They consist of numbers, variables (like $x$, $y$, etc.), and operation symbols (such as +, -, *, /). In the exercise provided, the expression $(4^x+1)(5^{x-1})$ includes both constants and variables. The constant here is 1, and the expressions involve powers of 4 and 5. Understanding how to set up expressions like these helps in evaluating complex problems more easily.
It's crucial to simplify algebraic expressions step by step. This often means substituting known values for variables or simplifying expressions using arithmetic rules. For instance, after finding that $x=5$, we substitute to get $(4^5+1)(5^4)$. Dealing with such expressions involves care, as calculations can be intricate and require logical sequencing of operations.

Exponentiation

Exponentiation is the mathematical operation that involves raising a base number to the power of an exponent. It's one of the fundamental operations in algebra and helps solve many common GRE math problems. In the given problem, you need to identify that $4^x=1,024$. Recognizing this requires knowledge that $4^5=1,024$, which suggests $x=5$.
Exponentiation can be seen across many mathematical scenarios. Common properties include:

• $a^m\times a^n=a^{m+n}$
• $(a^m{)}^n=a^{m\times n}$
• $a^0=1$, for all non-zero $a$

Applying these properties enables you to transform and simplify problems more effectively, making exponentiation a powerful tool for problem-solving.

Problem Solving

Problem-solving in mathematics often consists of understanding the problem, devising a plan, carrying out the plan, and then evaluating the solution. The original exercise is a great example of this structured approach.
First, we need to understand the equation $4^x=1,024$ and its solution, $x=5$. With this understanding, the next step is to substitute and simplify the given expression $(4^x+1)(5^{x-1})$. Through substitution, we arrive at $(1,025)(5^4)$.
Evaluating this further requires:

• Calculating $5^4=625$
• Multiplying by 1,025 to find the result

Lastly, the solution should be checked against provided options to ensure correctness. This clear, systematic approach is vital for successfully tackling GRE math questions.