Question

rewrite the expression using only positive exponents, and simplify. (assume that any variables in the expression are nonzero.) \\( x^8(4x^0 y^6)(8y)^0 \\)

evaluate the expression without using a calculator. (enter your answer in scientific notation.) \\( (9 \times 10^4)^2 \\) \\( \times 10 \\)

Explanation:

First Problem: Rewrite the expression \( x^{8}(4x^{0}y^{6})(8y)^{0} \) using only positive exponents, and simplify. (Assume that any variables in the expression are nonzero.)

Step 1: Recall the zero - exponent rule

The zero - exponent rule states that for any non - zero number \( a \), \( a^{0}=1 \). So we can simplify \( x^{0} \) and \( (8y)^{0} \) first. Since \( x
eq0 \) and \( y
eq0 \), \( x^{0} = 1 \) and \( (8y)^{0}=1 \).
The expression becomes \( x^{8}(4\times1\times y^{6})(1) \)

Step 2: Multiply the constant terms

Multiply the constant term 4 with the other constants (in this case, just 1 from \( (8y)^{0} \)). So \( 4\times1\times1 = 4 \)
The expression is now \( 4x^{8}y^{6} \)

Second Problem: Evaluate the expression \( (9\times10^{4})^{2} \) without using a calculator. (Enter your answer in scientific notation.)

Step 1: Use the power of a product rule

The power of a product rule states that \( (ab)^{n}=a^{n}b^{n} \). For \( (9\times10^{4})^{2} \), we have \( 9^{2}\times(10^{4})^{2} \)

Step 2: Calculate \( 9^{2} \) and \( (10^{4})^{2} \)

We know that \( 9^{2}=81 \). For \( (10^{4})^{2} \), we use the power of a power rule \( (a^{m})^{n}=a^{mn} \). So \( (10^{4})^{2}=10^{4\times2}=10^{8} \)

Step 3: Rewrite 81 in scientific notation

81 can be written as \( 8.1\times10^{1} \)

Step 4: Multiply the two scientific notation terms

When multiplying numbers in scientific notation \( a\times10^{m}\times b\times10^{n}=(a\times b)\times10^{m + n} \). Here, \( a = 8.1 \), \( b=1 \), \( m = 1 \) and \( n = 8 \)
So \( 8.1\times10^{1}\times10^{8}=8.1\times10^{1 + 8}=8.1\times10^{9} \) (We can also calculate \( 81\times10^{8}=8.1\times10\times10^{8}=8.1\times10^{9} \))

Answer:

s:

1. For the first expression: \( \boldsymbol{4x^{8}y^{6}} \)
2. For the second expression: \( \boldsymbol{8.1\times10^{9}} \)