Evaluate 10^0.2.

Step-by-step solution

1. 1

   Convert the decimal exponent to a fraction. Exponent rules are stated for fractions, so rewrite $0.2$ in that form:

   $0.2=\frac{2}{10}=\frac{1}{5}$

   Reducing to $\tfrac15$ matters — it reveals a unit fraction, which corresponds to a plain root with no extra power.

2. 2

   Apply the fractional exponent rule. For a positive base, $a^{1/n}=\sqrt[n]{a}$, so

   $10^{0.2}=10^{1/5}=\sqrt[5]{10}$

   This is the exact value: the number which, raised to the fifth power, gives $10$.

3. 3

   Estimate before computing. Since $1^{5}=1$ and $2^{5}=32$, the answer lies between $1$ and $2$, and much closer to $1.5$ than to $2$ because $1.5^{5}\approx7.6$ is already near $10$. That bracket rules out gross errors.

4. 4

   Give the decimal value.

   $\sqrt[5]{10}\approx1.5848932$

   Check by raising it back: $1.5848932^{5}=10.000000$ ✓ to seven figures, and the two computations $10^{0.2}$ and $10^{1/5}$ agree to machine precision ✓.

5. 5

   Note where this constant appears. $10^{0.2}$ is the ratio between consecutive one-third-octave frequency bands in acoustics, and $10^{1/10}\approx1.2589$ (its square root) is the factor corresponding to one decibel of power. Also worth remembering: $\left(10^{0.2}\right)^{5}=10^{1}=10$, so five of these factors multiply back to a clean decade.