1. What is the result of the expression: 1) 3^32 + - 27 + 1; 3) {/0.0081 - 16; 2) 812; 2. Solve the equation: 1

Зимний_Вечер

19

1.
a) To find the result of the expression \(3^{32} - 27 + 1\), we can apply the order of operations (also known as PEMDAS) to simplify it step by step.

First, let"s calculate \(3^{32}\). To do so, we raise 3 to the power of 32 using exponentiation. This gives us:
\[3^{32} = 1853020188851841.\]

Next, we subtract 27 from the result:
\[1853020188851841 - 27 = 1853020188851814.\]

Finally, we add 1 to obtain the final result:
\[1853020188851814 + 1 = 1853020188851815.\]

Therefore, the result of the expression \(3^{32} - 27 + 1\) is \(1853020188851815\).

b) To evaluate the expression \(\frac{1}{{0.0081}} - 16\), we first need to calculate the reciprocal of \(0.0081\). The reciprocal of a number \(a\) is obtained by dividing 1 by \(a\), so:
\[\frac{1}{{0.0081}} = 123.456790123456790.\]

Next, we subtract 16 from this result:
\[123.456790123456790 - 16 = 107.456790123456790.\]

Therefore, the value of \(\frac{1}{{0.0081}} - 16\) is \(107.456790123456790\).

c) The result of the expression \(812\) is simply \(812\).

2.
a) To solve the equation \(x^5 = 17\), we need to find the value of \(x\) that satisfies the equation. Taking the fifth root on both sides, we have:
\[x = \sqrt[5]{17}.\]

The fifth root of 17 is approximately \(1.825\). Therefore, the solution to the equation is \(x \approx 1.825\).

b) The equation \(2 = -2\) has no solution. This is because the left-hand side (LHS) is equal to 2, while the right-hand side (RHS) is equal to -2. Since 2 and -2 are not equal, there is no value of \(x\) that satisfies this equation.

c) The equation \(y = 27\) is already solved. The value of \(y\) is \(27\).

3.
To find the value of the expression \(17 - 73 - 17 + 73\), we simply need to perform the indicated operations:

\[17 - 73 - 17 + 73 = 0.\]

Therefore, the value of the expression \(17 - 73 - 17 + 73\) is \(0\).

4.
a) The function \(f(x) = 5x^0\) has a constant term of 5, since any number raised to the power of 0 equals 1. Therefore, the function \(f(x) = 5x^0\) is an even function.

b) The function \(f(x) = x + 2x\) can be simplified to \(f(x) = 3x\). Since the exponent of \(x\) is 1 in this case, the function \(f(x) = x + 2x\) is a linear function. Linear functions can have both even and odd characteristics depending on the coefficient of the leading term. In this case, since the coefficient is 3 (odd), the function \(f(x) = x + 2x\) is an odd function.

5.
To determine if the graph of the function \(y = x^2\) passes through the point \((-5, -125)\), we can substitute the x-coordinate (-5) into the equation and check if it satisfies the equation:

\[(-5)^2 = 25.\]

Since the y-coordinate given (-125) does not match the calculated value (25) when substituting the x-coordinate (-5), we can conclude that the graph of the function \(y = x^2\) does not pass through the point \((-5, -125)\).

6.
The equation \(0.3y - 2.4 = 0\) is a linear equation. To find the roots (values of \(y\) that make the equation true), we can solve it by isolating \(y\).

First, let"s add \(2.4\) to both sides of the equation:
\[0.3y = 2.4.\]

Next, divide both sides of the equation by \(0.3\) to solve for \(y\):
\[y = \frac{2.4}{0.3}.\]

Simplifying the right-hand side gives us:
\[y = 8.\]

Therefore, the root (solution) to the equation \(0.3y - 2.4 = 0\) is \(y = 8\).