Q. 1:x = x:64. What one number can replace x?

Answer

From the proportion 1:x = x:64 we write \(\frac{1}{x}=\frac{x}{64}\). Cross-multiplying gives
\[x^2=64.\]
Taking square roots yields
\[x=\pm 8.\]
If x must be positive, then
\[x=8\quad\text{or}\quad\boxed{8}.\]

Detailed Explanation

Problem: Solve the proportion \(1:x = x:64\)

Step 1 — Interpret the proportion

Rewrite the proportion as
\[
\frac{1}{x} = \frac{x}{64}.
\]

Step 2 — Clear denominators

Multiply both sides by \(64x\):
\[
64x\cdot\frac{1}{x} = 64x\cdot\frac{x}{64}.
\]
Simplifying each side gives
\[
64 = x^{2}.
\]

Step 3 — Solve the equation

\[
x^{2} = 64.
\]
Taking square roots,
\[
x = \pm\sqrt{64} = \pm 8.
\]

Step 4 — Verify

For \(x=8\):
\[
\frac{1}{8} = \frac{8}{64} = \frac{1}{8}.
\]
For \(x=-8\):
\[
\frac{1}{-8} = \frac{-8}{64} = -\frac{1}{8}.
\]
Both values satisfy the original proportion.

Answer

\[
x = 8 \quad\text{or}\quad x = -8.
\]
If only a positive value is required, choose \(x = 8\).

See full solution

Graph

Need help solving for x? Try our AI solver now.

AI solver

FAQs

What number can replace x in ( frac{1}{x} = frac{x}{64} )?

Cross-multiply: (1cdot 64 = x^2), so (x^2 = 64). Thus (x = 8) or (x = -8). If ratios require positive terms, pick (x = 8).

Why are there two solutions for (x)?

Because solving gives (x^2 = 64). A square equation yields two roots, the positive and negative square roots, (x = pm 8).

Can (x) be zero?

No. (x = 0) is invalid because the original expression has division by (x) (and by 64), so (x) cannot be zero.

How do you solve it using cross-multiplication?

Start with ( frac{1}{x} = frac{x}{64} ). Cross-multiply: (1cdot 64 = xcdot x), giving (64 = x^2). Solve (x = pm sqrt{64} = pm 8).

Is (x) the geometric mean of 1 and 64?

Yes: the positive solution (x=8) equals the geometric mean (sqrt{1cdot 64}=8). The negative root is not considered a geometric mean.

How can I check whether (x=8) or (x=-8) works?

Substitute: for (x=8), ( frac{1}{8} = frac{8}{64} = frac{1}{8}). For (x=-8), ( frac{1}{-8} = frac{-8}{64} = -frac{1}{8}). Both satisfy the equation.

If the problem uses colon notation 1:x = x:64, does it change anything?

No. The colon ratio (1:x = x:64) means the same as ( frac{1}{x} = frac{x}{64}). Solution and domain restrictions remain identical.

Try new AI picks today.
Choose yours right now

173,935+ happy customers
Math, Calculus, Geometry, etc.

Linear Algebra Solver Math AI Solver Calculus AI

Q. 1:x = x:64. What one number can replace x?

Answer

Detailed Explanation

Step 1 — Interpret the proportion

Step 2 — Clear denominators

Step 3 — Solve the equation

Step 4 — Verify

Answer

Graph

FAQs

What number can replace x in ( frac{1}{x} = frac{x}{64} )?

Why are there two solutions for (x)?

Can (x) be zero?

How do you solve it using cross-multiplication?

Is (x) the geometric mean of 1 and 64?

How can I check whether (x=8) or (x=-8) works?

If the problem uses colon notation 1:x = x:64, does it change anything?

Similar Questions

Go Premium to remove ads