Question

For what values of \(\mathrm{x}, 2^{\mathrm{x}} \times 5^{\mathrm{x}}\) ends in 5 ? (1) 0 (2) 1 (3) 2 (4) No value of \(x\)

Short answer

Answer: (4) No value of \(x\)

Step-by-step solution

Rewrite the given expression

We can rewrite \(2^{\mathrm{x}} \times 5^{\mathrm{x}}\) as \((2 \times 5)^{\mathrm{x}}\) which simplifies to \(10^{\mathrm{x}}\).

Analyze the options

We have four options to check for which values of \(x\), \(10^{\mathrm{x}}\) ends in 5. (1) If \(x = 0\): \(10^{\mathrm{0}} = 1\) (2) If \(x = 1\): \(10^{\mathrm{1}} = 10\) (3) If \(x = 2\): \(10^{\mathrm{2}} = 100\) (4) No value of \(x\) We can see that none of the values of \(x\) provided makes the expression \(10^{\mathrm{x}}\) end in 5.

Conclusion

Based on our analysis, none of the provided values of \(x\) make the given expression \(2^{\mathrm{x}} \times 5^{\mathrm{x}}\) to have a result that ends in 5. Therefore, the correct Answer is (4) No value of \(x\).

Key concepts

mathematical expressions

Mathematical expressions are combinations of numbers, variables, and operations (like addition and multiplication) that represent a mathematical object or relationship. When working with expressions like \(2^x \times 5^x\), understanding how to manipulate and rewrite them is crucial.
In this problem, the expression \(2^x \times 5^x\) is simplified to \((2 \times 5)^x = 10^x\). This step is important because it allows us to explore the properties of powers of ten, which are much easier to evaluate and understand.

• The expression \(2^x \times 5^x\) represents multiplying two bases, 2 and 5, both raised to the same power \(x\).
• Simplifying it to \((2 \times 5)^x\) helps us see it as a power of 10, which reveals more about the pattern of the number endings.
• By rewriting in simpler terms, identifying patterns such as the number endings becomes more straightforward.

Understanding how mathematical expressions can be rewritten and simplified is a fundamental skill in solving algebraic problems effectively.

number endings

The ending digit, or last digit, of a number is an important aspect that can reveal patterns and properties, especially in powers. When dealing with powers of 10 such as \(10^x\), the ending digit is influenced solely by the base number's last digit.
The exercise explores if \(10^x\) can end in 5, using specific values for \(x\):

• For any exponent \(x\), the ending digit of \(10^x\) follows a predictable pattern.
• Raising 10 to any positive integer power results in numbers ending in 0 (e.g., 10, 100, 1000).
• Thus, \(10^x\) will never end in 5 for any integer value of \(x\).

This pattern emerges from the nature of multiplying 10, where each multiplication by 10 effectively shifts the digits left by one position, appending a zero to the end. Understanding number endings helps predict outcomes of exponentiation without complete calculations.

powers of ten

Powers of ten represent numbers that multiply ten by itself a certain number of times. They are one of the most recognizable series in mathematics due to their clean and predictable pattern:

• \(10^0 = 1\)
• \(10^1 = 10\)
• \(10^2 = 100\)
• The pattern continues as each subsequent power of ten is simply a 1 followed by the corresponding number of zeros.

Working with powers of ten is crucial in understanding the effects of scaling numbers in mathematics and in everyday scenarios like money or measurement. They highlight exponential growth or shrinkage quickly and are pivotal in scientific notation.
In this exercise, we determined that powers of ten never end in 5, confirming that no \(x\) makes \(10^x\) end with the digit 5. This demonstrates how understanding the properties of powers of ten can help solve algebraic problems by revealing inevitable patterns.