Sure, let's solve each problem step-by-step:
---
### Problem 1: [tex]\( x^2 = 169 \)[/tex]
To solve for [tex]\( x \)[/tex], we need to find the square root of 169.
1. Take the square root of both sides:
[tex]\[
\sqrt{x^2} = \pm\sqrt{169}
\][/tex]
2. This yields two solutions:
[tex]\[
x = 13 \quad \text{and} \quad x = -13
\][/tex]
So the solutions for [tex]\( x \)[/tex] are [tex]\( 13 \)[/tex] and [tex]\( -13 \)[/tex].
### Problem 2: [tex]\( 9b^2 = 25 \)[/tex]
To solve for [tex]\( b \)[/tex], follow these steps:
1. Divide both sides by 9 to isolate [tex]\( b^2 \)[/tex]:
[tex]\[
b^2 = \frac{25}{9}
\][/tex]
2. Take the square root of both sides:
[tex]\[
\sqrt{b^2} = \pm\sqrt{\frac{25}{9}}
\][/tex]
3. Simplify the square root:
[tex]\[
b = \pm\frac{5}{3}
\][/tex]
So the solutions for [tex]\( b \)[/tex] are [tex]\( \frac{5}{3} \)[/tex] (approximately 1.6666666666666667) and [tex]\( -\frac{5}{3} \)[/tex] (approximately -1.6666666666666667).
### Problem 3: [tex]\( (x-2)^2 = 16 \)[/tex]
To solve for [tex]\( x \)[/tex], follow these steps:
1. Take the square root of both sides:
[tex]\[
\sqrt{(x-2)^2} = \pm\sqrt{16}
\][/tex]
2. This gives:
[tex]\[
x - 2 = \pm4
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[
x - 2 = 4 \implies x = 6
\][/tex]
[tex]\[
x - 2 = -4 \implies x = -2
\][/tex]
So the solutions for [tex]\( x \)[/tex] are [tex]\( 6 \)[/tex] and [tex]\( -2 \)[/tex].
### Problem 4: [tex]\( 2(t-3)^2 - 72 = 0 \)[/tex]
To solve for [tex]\( t \)[/tex], follow these steps:
1. Add 72 to both sides to isolate the squared term:
[tex]\[
2(t-3)^2 = 72
\][/tex]
2. Divide both sides by 2:
[tex]\[
(t-3)^2 = 36
\][/tex]
3. Take the square root of both sides:
[tex]\[
\sqrt{(t-3)^2} = \pm\sqrt{36}
\][/tex]
4. This gives:
[tex]\[
t - 3 = \pm6
\][/tex]
5. Solve for [tex]\( t \)[/tex]:
[tex]\[
t - 3 = 6 \implies t = 9
\][/tex]
[tex]\[
t - 3 = -6 \implies t = -3
\][/tex]
So the solutions for [tex]\( t \)[/tex] are [tex]\( 9 \)[/tex] and [tex]\( -3 \)[/tex].
---
In summary, the solutions are:
1. [tex]\( x = 13 \)[/tex] or [tex]\( x = -13 \)[/tex]
2. [tex]\( b = 1.6666666666666667 \)[/tex] or [tex]\( b = -1.6666666666666667 \)[/tex]
3. [tex]\( x = 6 \)[/tex] or [tex]\( x = -2 \)[/tex]
4. [tex]\( t = 9 \)[/tex] or [tex]\( t = -3 \)[/tex]
