# Powers roots and very small numbers

Subject: Deep Learning

Exam: TEE

Reader: https://deep-learning-study-room.vercel.app/#tee/read/2/tee-c2-s4

Canonical: https://deep-learning-study-room.vercel.app/read/dl/tee/section/tee-c2-s4/

### 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.



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



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.](https://deep-learning-study-room.vercel.app/assets/figures/v_powers.svg)

Diagram: Count the three rows of three cells. Squaring builds nine; the square root returns three.. Labels: A square and its square root | A square and its square root | Three rows of three cells make nine cells. | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3² = 3 × 3 = 9 | √9 = 3 | The root returns the side length. | (−2)² = (−2) × (−2) = 4. Keep the minus sign inside the brackets.

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.



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



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.



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



### 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.



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



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.



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



### 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.



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



### 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.



$$
q^2=q\times q
$$





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



**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.**



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



**Step 2: subtract the result from 1.**



$$
1-0.81=0.19
$$



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



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