Find x if 132x = 70 eight.

Analysis

The question asks us to find the value of the base $x$ in the equation $132_x = 70_8$. This is a problem involving Number Bases. To solve this, we need to convert both sides of the equation into base 10 (decimal) and then solve for $x$.

Concept

To convert a number from any base $x$ to base 10, we multiply each digit by $x$ raised to the power of its position (starting from 0 on the right).

For a three-digit number $abc_x$: $abc_x = (a \times x^2) + (b \times x^1) + (c \times x^0)$

Solution

Step 1: Convert the left side ($132_x$) to base 10. $132_x = (1 \times x^2) + (3 \times x^1) + (2 \times x^0)$ $132_x = x^2 + 3x + 2$

Step 2: Convert the right side ($70_8$) to base 10. $70_8 = (7 \times 8^1) + (0 \times 8^0)$ $70_8 = (7 \times 8) + (0 \times 1)$ $70_8 = 56$

Step 3: Equate both sides and solve the quadratic equation. $x^2 + 3x + 2 = 56$ $x^2 + 3x + 2 - 56 = 0$ $x^2 + 3x - 54 = 0$

Step 4: Factorize the quadratic equation. We need two numbers that multiply to $-54$ and add up to $+3$. Those numbers are $+9$ and $-6$. $(x + 9)(x - 6) = 0$

This gives us two possible values for $x$: $x = -9$ or $x = 6$

Since a number base must be a positive integer, we discard $-9$. Therefore, $x = 6$.

Conclusion

The value of $x$ is 6. This matches option C.

Exam Tip

When dealing with number bases, always remember that the base must be greater than any individual digit in the number. For example, in $132_x$, $x$ must be greater than 3. Since our result is 6, it satisfies this condition!