Solve a^{2^{-1}}+8^{3^-1}+625^4^-1+(1/4)^-2

$$a ^ { 2 ^ { - 1 } } + 8 ^ { 3 ^ { - 1 } } + 625 ^ { 4 ^ { - 1 } } + ( \frac { 1 } { 4 } ) ^ { - 2 }$$

Evaluate

$$\sqrt{a}+23$$

Solution Steps

Calculate 2 to the power of -1 and get \frac{1}{2}.

$$a^{\frac{1}{2}}+8^{3^{-1}}+625^{4^{-1}}+\left(\frac{1}{4}\right)^{-2}$$

Calculate 3 to the power of -1 and get \frac{1}{3}.

$$a^{\frac{1}{2}}+8^{\frac{1}{3}}+625^{4^{-1}}+\left(\frac{1}{4}\right)^{-2}$$

Calculate 8 to the power of \frac{1}{3} and get 2.

$$a^{\frac{1}{2}}+2+625^{4^{-1}}+\left(\frac{1}{4}\right)^{-2}$$

Calculate 4 to the power of -1 and get \frac{1}{4}.

$$a^{\frac{1}{2}}+2+625^{\frac{1}{4}}+\left(\frac{1}{4}\right)^{-2}$$

Calculate 625 to the power of \frac{1}{4} and get 5.

$$a^{\frac{1}{2}}+2+5+\left(\frac{1}{4}\right)^{-2}$$

Add 2 and 5 to get 7.

$$a^{\frac{1}{2}}+7+\left(\frac{1}{4}\right)^{-2}$$

Calculate \frac{1}{4} to the power of -2 and get 16.

$$a^{\frac{1}{2}}+7+16$$

Add 7 and 16 to get 23.

$$a^{\frac{1}{2}}+23$$

Differentiate w.r.t. a

$$\frac{1}{2\sqrt{a}}$$

Steps Using Derivative Rule for Sum

Calculate 2 to the power of -1 and get \frac{1}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{\frac{1}{2}}+8^{3^{-1}}+625^{4^{-1}}+\left(\frac{1}{4}\right)^{-2})$$

Calculate 3 to the power of -1 and get \frac{1}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{\frac{1}{2}}+8^{\frac{1}{3}}+625^{4^{-1}}+\left(\frac{1}{4}\right)^{-2})$$

Calculate 8 to the power of \frac{1}{3} and get 2.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{\frac{1}{2}}+2+625^{4^{-1}}+\left(\frac{1}{4}\right)^{-2})$$

Calculate 4 to the power of -1 and get \frac{1}{4}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{\frac{1}{2}}+2+625^{\frac{1}{4}}+\left(\frac{1}{4}\right)^{-2})$$

Calculate 625 to the power of \frac{1}{4} and get 5.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{\frac{1}{2}}+2+5+\left(\frac{1}{4}\right)^{-2})$$

Add 2 and 5 to get 7.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{\frac{1}{2}}+7+\left(\frac{1}{4}\right)^{-2})$$

Calculate \frac{1}{4} to the power of -2 and get 16.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{\frac{1}{2}}+7+16)$$

Add 7 and 16 to get 23.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{\frac{1}{2}}+23)$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$\frac{1}{2}a^{\frac{1}{2}-1}$$

Subtract 1 from \frac{1}{2}.

$$\frac{1}{2}a^{-\frac{1}{2}}$$