Since 1000 can be expressed as 10^3, we have 10^x = 10^3, thus x = 3.

Since 1000 can be expressed as 10^3, we have 10^(x+2) = 10^3, thus x + 2 = 3, leading to x = 1.

Since 64 can be expressed as 4^3, we have 4^(x-1) = 4^3, thus x - 1 = 3, leading to x = 4.

Since 1/49 can be expressed as 7^(-2), we have 7^x = 7^(-2), thus x = -2.

The exponent zero indicates the multiplicative identity, meaning any non-zero number raised to the power of zero equals one.

Using the property of exponents, a^3 * a^(-2) = a^(3 - 2) = a^1, hence x = 1.

Substituting a = 2, we have 2^3 * b^2 = 64, which simplifies to 8b^2 = 64. Thus, b^2 = 8, leading to b = 4.

If a = b, then a^x = b^y implies x must equal y for the equality to hold.

We have x = 8 and y = 4. Thus, x/y = 8/4 = 2.

Since 125 can be expressed as 5^3, we have 5^(x+1) = 5^3, thus x + 1 = 3, leading to x = 2.

Exponents indicate how many times a base is multiplied by itself, which is fundamental in understanding powers in mathematics.

Using the property of exponents (a^m)^n = a^(m*n), we have (2^3)^2 = 2^(3*2) = 2^6.

Using the power of a product property, we have (x^2 * y^3)^(2) = x^(2*2) * y^(3*2) = x^4 * y^6.

Calculating each term, we have 5^0 = 1, 5^1 = 5, and 5^2 = 25. Therefore, 1 + 5 + 25 = 31.

Using the property of exponents, we have 5^(2 + 3 - 4) = 5^1 = 5.

Using the property of exponents that states a^m * a^n = a^(m+n), we combine the exponents: 2x + (x + 1) = 3x + 1.

Using the power of a product property, (a*b)^n = a^n * b^n, we get (x^3)^2 * (y^2)^2 = x^6 * y^4.

Using the power of a power property, (a^m)^n = a^(m*n), we get (2^3)^2 = 2^(3*2) = 2^6.

Using the property of exponents, we have 2^(x+1 - (x-1)) = 2^(x+1-x+1) = 2^2.

Since 16 can be expressed as 4^2, we have 4^(x+1) = 4^2, leading to x + 1 = 2, thus x = 1.

The statement a^m + a^n = a^(m+n) is incorrect; addition of exponents does not apply in this manner.