Questions: 1/3+[((-2/3)^2(2/3)^4)^2:(-2/3)^10](3/7^2)+1/4-2/3

Transcript text: $\left\{\frac{1}{3}+\left[\left(-\frac{2}{3}\right)^{2} \cdot\left(\frac{2}{3}\right)^{4}\right]^{2}:\left(-\frac{2}{3}\right)^{10}\right\} \cdot\left(\frac{3}{7^{2}}\right)+\frac{1}{4}-\frac{2}{3}$

Solution

Solution Steps

To solve the given expression, we need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). First, evaluate the innermost expressions and exponents, then proceed with multiplication and division, and finally handle addition and subtraction.

Step 1: Evaluate Inner Expressions

First, we calculate the inner expressions. We have: \[ \left(-\frac{2}{3}\right)^{2} \cdot \left(\frac{2}{3}\right)^{4} = 0.08779149519890259 \] Next, we square this result: \[ \left(0.08779149519890259\right)^{2} = 0.007707346629258936 \]

Step 2: Division

Now, we divide this squared result by $\left(-\frac{2}{3}\right)^{10}$: \[ \frac{0.007707346629258936}{\left(-\frac{2}{3}\right)^{10}} = 0.4444444444444445 \]

Step 3: Combine All Parts

We then combine all parts of the expression: \[ \left\{\frac{1}{3} + 0.4444444444444445\right\} \cdot \left(\frac{3}{7^{2}}\right) + \frac{1}{4} - \frac{2}{3} \] Calculating this gives: \[ \left(0.3333333333333333 + 0.4444444444444445\right) \cdot 0.061224489795918366 + 0.25 - 0.6666666666666666 = -0.369047619047619 \]