Deep Learning · MT2 · section

Powers roots and very small numbers

Mathematics revision

1. A power counts repeated multiplication

In 2³, 2 is the base and 3 is the exponent. For a positive whole-number exponent, multiply that many copies of the base. It does not mean 2 × 3.

$2^3=2\times2\times2=8$ LaTeX: 2^3=2\times2\times2=8

The exponent 2 is called a square: 3² = 3 × 3 = 9. Three rows of three cells make nine cells.

Count the three rows of three cells. Squaring builds nine; the square root returns three. Tap the diagram to enlarge.

2. A negative number must stay inside its brackets

Squaring negative 2 means multiplying negative 2 by negative 2. The equal signs of the factors give a positive result.

$(-2)^2=(-2)(-2)=4$ LaTeX: (-2)^2=(-2)(-2)=4

Without brackets, −2² means first calculate 2², then apply the minus sign: −(4) = −4. Enter brackets when the entire negative number is the base.

For decimals, use the same rule. Negative 0.3 squared is positive 0.09 because 3/10 multiplied by 3/10 is 9/100.

$\begin{aligned}(-0.3)^2&=(-0.3)(-0.3)\\&=0.09\end{aligned}$ LaTeX: \begin{aligned}(-0.3)^2&=(-0.3)(-0.3)\\&=0.09\end{aligned}

3. A square root asks which nonnegative number was squared

Because 3 × 3 = 9, the principal square root of 9 is 3. Although (−3)² also equals 9, the symbol √9 by itself means the nonnegative root, 3.

$\begin{aligned}\sqrt9&=3\\\sqrt{0.09}&=0.3\end{aligned}$ LaTeX: \begin{aligned}\sqrt9&=3\\\sqrt{0.09}&=0.3\end{aligned}

The absolute value of a number is its distance from zero: |−4| = 4 and |4| = 4. Squaring and taking a square root returns this nonnegative size.

$\sqrt{(-4)^2}=\sqrt{16}=4$ LaTeX: \sqrt{(-4)^2}=\sqrt{16}=4

4. Zero and negative exponents follow a pattern

Start with powers of 10. Each time the exponent decreases by 1, divide the previous value by 10: 1000 → 100 → 10 → 1 → 0.1 → 0.01. This explains both the zero power and the negative powers.

Power	Calculation	Value
10³	10 × 10 × 10	1000
10²	1000 ÷ 10	100
10¹	100 ÷ 10	10
10⁰	10 ÷ 10	1
10⁻¹	1 ÷ 10	0.1
10⁻²	0.1 ÷ 10	0.01
10⁻³	0.01 ÷ 10	0.001

For a nonzero base, power zero is 1. A negative exponent means the reciprocal of the corresponding positive power. It does not mean a negative answer.

$\begin{aligned}10^{-3}&=\frac1{10^3}\\&=\frac1{1000}\\&=0.001\end{aligned}$ LaTeX: \begin{aligned}10^{-3}&=\frac1{10^3}\\&=\frac1{1000}\\&=0.001\end{aligned}

5. Tiny scientific notation is still an ordinary number

The notation 1 × 10⁻⁸ equals 0.00000001. Calculators may display it as 1E−8. In optimizer formulas, ε, pronounced “epsilon”, often names a small positive constant added to a denominator. Read its supplied value before calculating.

Question. Calculate 1 − 0.9².

Given

Expression: 1−0.9².

To find

Evaluate the power first, then the subtraction.

Formula — which one and why

A square before subtraction. The exponent applies to 0.9. Squaring is repeated multiplication; perform it before subtracting the result from 1.

q denotes the number being squared. Parentheses would be required to square the whole difference.

$q^2=q\times q$ LaTeX: q^2=q\times q

$1-q^2=1-(q\times q)$ LaTeX: 1-q^2=1-(q\times q)

Yaad rakhne ka cue: Jis number pe power hai, sirf uska square; subtraction baad mein.

Solution — step by step

Step 1: square only the 0.9.

$0.9^2=0.9(0.9)=0.81$ LaTeX: 0.9^2=0.9(0.9)=0.81

Step 2: subtract the result from 1.

$1-0.81=0.19$ LaTeX: 1-0.81=0.19

Final answer: 0.19. The square acts on 0.9, not on the entire difference.

$\begin{aligned}1-0.9^2&=1-0.81\\&=0.19\end{aligned}$ LaTeX: \begin{aligned}1-0.9^2&=1-0.81\\&=0.19\end{aligned}